let I be BinOp of [.0,1.]; :: thesis: ( I is satisfying_(OP) implies ( I is satisfying_(I3) & I is satisfying_(I4) & I is satisfying_(NC) & I is satisfying_(LB) & I is satisfying_(RB) & I is satisfying_(IP) ) )
assume a1: I is satisfying_(OP) ; :: thesis: ( I is satisfying_(I3) & I is satisfying_(I4) & I is satisfying_(NC) & I is satisfying_(LB) & I is satisfying_(RB) & I is satisfying_(IP) )
A2: 0 in [.0,1.] by XXREAL_1:1;
A3: 1 in [.0,1.] by XXREAL_1:1;
T3: I is satisfying_(LB)
proof
let y be Element of [.0,1.]; :: according to FUZIMPL1:def 26 :: thesis: I . (K38(),y) = 1
0 <= y by XXREAL_1:1;
hence I . (K38(),y) = 1 by a1, A2; :: thesis: verum
end;
I is satisfying_(RB)
proof
let x be Element of [.0,1.]; :: according to FUZIMPL1:def 27 :: thesis: I . (x,1) = 1
x <= 1 by XXREAL_1:1;
hence I . (x,1) = 1 by a1, A3; :: thesis: verum
end;
hence ( I is satisfying_(I3) & I is satisfying_(I4) & I is satisfying_(NC) & I is satisfying_(LB) & I is satisfying_(RB) & I is satisfying_(IP) ) by T3, a1; :: thesis: verum