let L be non empty LattStr ; :: thesis: ( ( for v0 being Element of L holds v0 "/\" v0 = v0 ) & ( for v2, v1, v0 being Element of L holds (v0 "/\" v1) "/\" v2 = v0 "/\" (v1 "/\" v2) ) & ( for v1, v0 being Element of L holds v0 "/\" v1 = v1 "/\" v0 ) & ( for v0 being Element of L holds v0 "\/" v0 = v0 ) & ( for v2, v1, v0 being Element of L holds (v0 "\/" v1) "\/" v2 = v0 "\/" (v1 "\/" v2) ) & ( for v0, v2, v1 being Element of L holds (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) ) & ( for v0, v2, v1 being Element of L holds v0 "/\" (v1 "\/" v2) = (v0 "/\" v1) "\/" (v0 "/\" v2) ) implies for v1, v2, v3 being Element of L holds v1 "\/" (v2 "/\" v3) = (v1 "\/" v2) "/\" (v1 "\/" v3) )
assume A2: for v0 being Element of L holds v0 "/\" v0 = v0 ; :: thesis: ( ex v2, v1, v0 being Element of L st not (v0 "/\" v1) "/\" v2 = v0 "/\" (v1 "/\" v2) or ex v1, v0 being Element of L st not v0 "/\" v1 = v1 "/\" v0 or ex v0 being Element of L st not v0 "\/" v0 = v0 or ex v2, v1, v0 being Element of L st not (v0 "\/" v1) "\/" v2 = v0 "\/" (v1 "\/" v2) or ex v0, v2, v1 being Element of L st not (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) or ex v0, v2, v1 being Element of L st not v0 "/\" (v1 "\/" v2) = (v0 "/\" v1) "\/" (v0 "/\" v2) or for v1, v2, v3 being Element of L holds v1 "\/" (v2 "/\" v3) = (v1 "\/" v2) "/\" (v1 "\/" v3) )
assume A3: for v2, v1, v0 being Element of L holds (v0 "/\" v1) "/\" v2 = v0 "/\" (v1 "/\" v2) ; :: thesis: ( ex v1, v0 being Element of L st not v0 "/\" v1 = v1 "/\" v0 or ex v0 being Element of L st not v0 "\/" v0 = v0 or ex v2, v1, v0 being Element of L st not (v0 "\/" v1) "\/" v2 = v0 "\/" (v1 "\/" v2) or ex v0, v2, v1 being Element of L st not (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) or ex v0, v2, v1 being Element of L st not v0 "/\" (v1 "\/" v2) = (v0 "/\" v1) "\/" (v0 "/\" v2) or for v1, v2, v3 being Element of L holds v1 "\/" (v2 "/\" v3) = (v1 "\/" v2) "/\" (v1 "\/" v3) )
assume A4: for v1, v0 being Element of L holds v0 "/\" v1 = v1 "/\" v0 ; :: thesis: ( ex v0 being Element of L st not v0 "\/" v0 = v0 or ex v2, v1, v0 being Element of L st not (v0 "\/" v1) "\/" v2 = v0 "\/" (v1 "\/" v2) or ex v0, v2, v1 being Element of L st not (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) or ex v0, v2, v1 being Element of L st not v0 "/\" (v1 "\/" v2) = (v0 "/\" v1) "\/" (v0 "/\" v2) or for v1, v2, v3 being Element of L holds v1 "\/" (v2 "/\" v3) = (v1 "\/" v2) "/\" (v1 "\/" v3) )
assume A5: for v0 being Element of L holds v0 "\/" v0 = v0 ; :: thesis: ( ex v2, v1, v0 being Element of L st not (v0 "\/" v1) "\/" v2 = v0 "\/" (v1 "\/" v2) or ex v0, v2, v1 being Element of L st not (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) or ex v0, v2, v1 being Element of L st not v0 "/\" (v1 "\/" v2) = (v0 "/\" v1) "\/" (v0 "/\" v2) or for v1, v2, v3 being Element of L holds v1 "\/" (v2 "/\" v3) = (v1 "\/" v2) "/\" (v1 "\/" v3) )
assume A6: for v2, v1, v0 being Element of L holds (v0 "\/" v1) "\/" v2 = v0 "\/" (v1 "\/" v2) ; :: thesis: ( ex v0, v2, v1 being Element of L st not (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) or ex v0, v2, v1 being Element of L st not v0 "/\" (v1 "\/" v2) = (v0 "/\" v1) "\/" (v0 "/\" v2) or for v1, v2, v3 being Element of L holds v1 "\/" (v2 "/\" v3) = (v1 "\/" v2) "/\" (v1 "\/" v3) )
assume A7: for v0, v2, v1 being Element of L holds (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) ; :: thesis: ( ex v0, v2, v1 being Element of L st not v0 "/\" (v1 "\/" v2) = (v0 "/\" v1) "\/" (v0 "/\" v2) or for v1, v2, v3 being Element of L holds v1 "\/" (v2 "/\" v3) = (v1 "\/" v2) "/\" (v1 "\/" v3) )
A9: for v0, v2, v1 being Element of L holds (v0 "\/" v1) "/\" (v0 "\/" (v1 "/\" v2)) = v0 "\/" (v1 "/\" v2)
proof
let v0, v2, v1 be Element of L; :: thesis: (v0 "\/" v1) "/\" (v0 "\/" (v1 "/\" v2)) = v0 "\/" (v1 "/\" v2)
(v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = (v0 "\/" v1) "/\" (v0 "\/" (v1 "/\" v2)) by A4;
hence (v0 "\/" v1) "/\" (v0 "\/" (v1 "/\" v2)) = v0 "\/" (v1 "/\" v2) by A7; :: thesis: verum
end;
assume A10: for v0, v2, v1 being Element of L holds v0 "/\" (v1 "\/" v2) = (v0 "/\" v1) "\/" (v0 "/\" v2) ; :: thesis: for v1, v2, v3 being Element of L holds v1 "\/" (v2 "/\" v3) = (v1 "\/" v2) "/\" (v1 "\/" v3)
A15: for v102, v101 being Element of L holds v101 "/\" v102 = v101 "/\" (v101 "/\" v102)
proof
let v102, v101 be Element of L; :: thesis: v101 "/\" v102 = v101 "/\" (v101 "/\" v102)
v101 "/\" v101 = v101 by A2;
hence v101 "/\" v102 = v101 "/\" (v101 "/\" v102) by A3; :: thesis: verum
end;
A20: for v102, v101 being Element of L holds v101 "\/" v102 = v101 "\/" (v101 "\/" v102)
proof
let v102, v101 be Element of L; :: thesis: v101 "\/" v102 = v101 "\/" (v101 "\/" v102)
v101 "\/" v101 = v101 by A5;
hence v101 "\/" v102 = v101 "\/" (v101 "\/" v102) by A6; :: thesis: verum
end;
A24: for v2, v0, v1 being Element of L holds (v1 "/\" v0) "\/" (v0 "/\" v2) = v0 "/\" (v1 "\/" v2)
proof
let v2, v0, v1 be Element of L; :: thesis: (v1 "/\" v0) "\/" (v0 "/\" v2) = v0 "/\" (v1 "\/" v2)
v0 "/\" v1 = v1 "/\" v0 by A4;
hence (v1 "/\" v0) "\/" (v0 "/\" v2) = v0 "/\" (v1 "\/" v2) by A10; :: thesis: verum
end;
A28: for v102, v0, v2, v1 being Element of L holds (v0 "/\" (v1 "\/" v2)) "\/" v102 = (v0 "/\" v1) "\/" ((v0 "/\" v2) "\/" v102)
proof
let v102, v0, v2, v1 be Element of L; :: thesis: (v0 "/\" (v1 "\/" v2)) "\/" v102 = (v0 "/\" v1) "\/" ((v0 "/\" v2) "\/" v102)
(v0 "/\" v1) "\/" (v0 "/\" v2) = v0 "/\" (v1 "\/" v2) by A10;
hence (v0 "/\" (v1 "\/" v2)) "\/" v102 = (v0 "/\" v1) "\/" ((v0 "/\" v2) "\/" v102) by A6; :: thesis: verum
end;
A32: for v102, v1, v100 being Element of L holds (v100 "/\" v1) "\/" (v100 "/\" v102) = v100 "/\" ((v100 "/\" v1) "\/" v102)
proof
let v102, v1, v100 be Element of L; :: thesis: (v100 "/\" v1) "\/" (v100 "/\" v102) = v100 "/\" ((v100 "/\" v1) "\/" v102)
v100 "/\" (v100 "/\" v1) = v100 "/\" v1 by A15;
hence (v100 "/\" v1) "\/" (v100 "/\" v102) = v100 "/\" ((v100 "/\" v1) "\/" v102) by A10; :: thesis: verum
end;
A35: for v0, v2, v1 being Element of L holds v0 "/\" (v1 "\/" v2) = v0 "/\" ((v0 "/\" v1) "\/" v2)
proof
let v0, v2, v1 be Element of L; :: thesis: v0 "/\" (v1 "\/" v2) = v0 "/\" ((v0 "/\" v1) "\/" v2)
(v0 "/\" v1) "\/" (v0 "/\" v2) = v0 "/\" (v1 "\/" v2) by A10;
hence v0 "/\" (v1 "\/" v2) = v0 "/\" ((v0 "/\" v1) "\/" v2) by A32; :: thesis: verum
end;
A39: for v1, v101, v100 being Element of L holds (v100 "/\" v101) "\/" (v100 "/\" v1) = v100 "/\" (v101 "\/" (v100 "/\" v1))
proof
let v1, v101, v100 be Element of L; :: thesis: (v100 "/\" v101) "\/" (v100 "/\" v1) = v100 "/\" (v101 "\/" (v100 "/\" v1))
v100 "/\" (v100 "/\" v1) = v100 "/\" v1 by A15;
hence (v100 "/\" v101) "\/" (v100 "/\" v1) = v100 "/\" (v101 "\/" (v100 "/\" v1)) by A10; :: thesis: verum
end;
A42: for v0, v2, v1 being Element of L holds v0 "/\" (v1 "\/" v2) = v0 "/\" (v1 "\/" (v0 "/\" v2))
proof
let v0, v2, v1 be Element of L; :: thesis: v0 "/\" (v1 "\/" v2) = v0 "/\" (v1 "\/" (v0 "/\" v2))
(v0 "/\" v1) "\/" (v0 "/\" v2) = v0 "/\" (v1 "\/" v2) by A10;
hence v0 "/\" (v1 "\/" v2) = v0 "/\" (v1 "\/" (v0 "/\" v2)) by A39; :: thesis: verum
end;
A46: for v102, v1, v100 being Element of L holds (v100 "\/" v1) "/\" (v100 "\/" ((v100 "\/" v1) "/\" v102)) = v100 "\/" ((v100 "\/" v1) "/\" v102)
proof
let v102, v1, v100 be Element of L; :: thesis: (v100 "\/" v1) "/\" (v100 "\/" ((v100 "\/" v1) "/\" v102)) = v100 "\/" ((v100 "\/" v1) "/\" v102)
v100 "\/" (v100 "\/" v1) = v100 "\/" v1 by A20;
hence (v100 "\/" v1) "/\" (v100 "\/" ((v100 "\/" v1) "/\" v102)) = v100 "\/" ((v100 "\/" v1) "/\" v102) by A9; :: thesis: verum
end;
A49: for v2, v1, v0 being Element of L holds (v0 "\/" v1) "/\" (v0 "\/" v2) = v0 "\/" ((v0 "\/" v1) "/\" v2)
proof
let v2, v1, v0 be Element of L; :: thesis: (v0 "\/" v1) "/\" (v0 "\/" v2) = v0 "\/" ((v0 "\/" v1) "/\" v2)
(v0 "\/" v1) "/\" (v0 "\/" ((v0 "\/" v1) "/\" v2)) = (v0 "\/" v1) "/\" (v0 "\/" v2) by A42;
hence (v0 "\/" v1) "/\" (v0 "\/" v2) = v0 "\/" ((v0 "\/" v1) "/\" v2) by A46; :: thesis: verum
end;
A53: for v2, v1, v101 being Element of L holds (v101 "\/" v1) "/\" (((v101 "\/" v1) "/\" v101) "\/" (v1 "/\" v2)) = v101 "\/" (v1 "/\" v2)
proof
let v2, v1, v101 be Element of L; :: thesis: (v101 "\/" v1) "/\" (((v101 "\/" v1) "/\" v101) "\/" (v1 "/\" v2)) = v101 "\/" (v1 "/\" v2)
(v101 "\/" v1) "/\" (v101 "\/" (v1 "/\" v2)) = v101 "\/" (v1 "/\" v2) by A9;
hence (v101 "\/" v1) "/\" (((v101 "\/" v1) "/\" v101) "\/" (v1 "/\" v2)) = v101 "\/" (v1 "/\" v2) by A35; :: thesis: verum
end;
A56: for v2, v1, v0 being Element of L holds (v0 "\/" v1) "/\" ((v0 "/\" (v0 "\/" v1)) "\/" (v1 "/\" v2)) = v0 "\/" (v1 "/\" v2)
proof
let v2, v1, v0 be Element of L; :: thesis: (v0 "\/" v1) "/\" ((v0 "/\" (v0 "\/" v1)) "\/" (v1 "/\" v2)) = v0 "\/" (v1 "/\" v2)
(v0 "\/" v1) "/\" v0 = v0 "/\" (v0 "\/" v1) by A4;
hence (v0 "\/" v1) "/\" ((v0 "/\" (v0 "\/" v1)) "\/" (v1 "/\" v2)) = v0 "\/" (v1 "/\" v2) by A53; :: thesis: verum
end;
A58: for v2, v1, v0 being Element of L holds (v0 "\/" v1) "/\" ((v0 "/\" v0) "\/" ((v0 "/\" v1) "\/" (v1 "/\" v2))) = v0 "\/" (v1 "/\" v2)
proof
let v2, v1, v0 be Element of L; :: thesis: (v0 "\/" v1) "/\" ((v0 "/\" v0) "\/" ((v0 "/\" v1) "\/" (v1 "/\" v2))) = v0 "\/" (v1 "/\" v2)
(v0 "/\" (v0 "\/" v1)) "\/" (v1 "/\" v2) = (v0 "/\" v0) "\/" ((v0 "/\" v1) "\/" (v1 "/\" v2)) by A28;
hence (v0 "\/" v1) "/\" ((v0 "/\" v0) "\/" ((v0 "/\" v1) "\/" (v1 "/\" v2))) = v0 "\/" (v1 "/\" v2) by A56; :: thesis: verum
end;
A60: for v2, v1, v0 being Element of L holds (v0 "\/" v1) "/\" (v0 "\/" ((v0 "/\" v1) "\/" (v1 "/\" v2))) = v0 "\/" (v1 "/\" v2)
proof
let v2, v1, v0 be Element of L; :: thesis: (v0 "\/" v1) "/\" (v0 "\/" ((v0 "/\" v1) "\/" (v1 "/\" v2))) = v0 "\/" (v1 "/\" v2)
v0 "/\" v0 = v0 by A2;
hence (v0 "\/" v1) "/\" (v0 "\/" ((v0 "/\" v1) "\/" (v1 "/\" v2))) = v0 "\/" (v1 "/\" v2) by A58; :: thesis: verum
end;
A62: for v1, v2, v0 being Element of L holds (v0 "\/" v1) "/\" (v0 "\/" (v1 "/\" (v0 "\/" v2))) = v0 "\/" (v1 "/\" v2)
proof
let v1, v2, v0 be Element of L; :: thesis: (v0 "\/" v1) "/\" (v0 "\/" (v1 "/\" (v0 "\/" v2))) = v0 "\/" (v1 "/\" v2)
(v0 "/\" v1) "\/" (v1 "/\" v2) = v1 "/\" (v0 "\/" v2) by A24;
hence (v0 "\/" v1) "/\" (v0 "\/" (v1 "/\" (v0 "\/" v2))) = v0 "\/" (v1 "/\" v2) by A60; :: thesis: verum
end;
A64: for v1, v2, v0 being Element of L holds v0 "\/" (v1 "/\" (v0 "\/" v2)) = v0 "\/" (v1 "/\" v2)
proof
let v1, v2, v0 be Element of L; :: thesis: v0 "\/" (v1 "/\" (v0 "\/" v2)) = v0 "\/" (v1 "/\" v2)
(v0 "\/" v1) "/\" (v0 "\/" (v1 "/\" (v0 "\/" v2))) = v0 "\/" (v1 "/\" (v0 "\/" v2)) by A9;
hence v0 "\/" (v1 "/\" (v0 "\/" v2)) = v0 "\/" (v1 "/\" v2) by A62; :: thesis: verum
end;
A66: for v1, v2, v0 being Element of L holds v0 "\/" ((v0 "\/" v2) "/\" v1) = v0 "\/" (v1 "/\" v2)
proof
let v1, v2, v0 be Element of L; :: thesis: v0 "\/" ((v0 "\/" v2) "/\" v1) = v0 "\/" (v1 "/\" v2)
v1 "/\" (v0 "\/" v2) = (v0 "\/" v2) "/\" v1 by A4;
hence v0 "\/" ((v0 "\/" v2) "/\" v1) = v0 "\/" (v1 "/\" v2) by A64; :: thesis: verum
end;
A69: for v2, v1, v0 being Element of L holds (v0 "\/" v1) "/\" (v0 "\/" v2) = v0 "\/" (v2 "/\" v1)
proof
let v2, v1, v0 be Element of L; :: thesis: (v0 "\/" v1) "/\" (v0 "\/" v2) = v0 "\/" (v2 "/\" v1)
v0 "\/" ((v0 "\/" v1) "/\" v2) = (v0 "\/" v1) "/\" (v0 "\/" v2) by A49;
hence (v0 "\/" v1) "/\" (v0 "\/" v2) = v0 "\/" (v2 "/\" v1) by A66; :: thesis: verum
end;
let v1, v2, v3 be Element of L; :: thesis: v1 "\/" (v2 "/\" v3) = (v1 "\/" v2) "/\" (v1 "\/" v3)
v1 "\/" (v2 "/\" v3) = (v1 "\/" v3) "/\" (v1 "\/" v2) by A69;
hence v1 "\/" (v2 "/\" v3) = (v1 "\/" v2) "/\" (v1 "\/" v3) by A4; :: thesis: verum