deffunc H1() -> set = {0,1,2};
deffunc H2( Element of H1(), Element of H1()) -> Element of {0,1,2} = OpEx2 ($1,$2);
ex f being BinOp of H1() st
for x, y being Element of H1() holds f . (x,y) = H2(x,y) from BINOP_1:sch 4();
 then consider f being BinOp of H1() such that
A1: for x, y being Element of H1() holds f . (x,y) = H2(x,y) ;
take f ; :: thesis: for x, y being Element of {0,1,2} holds
( ( ( x = 1 or y = 1 ) implies f . (x,y) = 1 ) & ( x <> 1 & y <> 1 implies f . (x,y) = min (x,y) ) )
let x, y be Element of {0,1,2}; :: thesis: ( ( ( x = 1 or y = 1 ) implies f . (x,y) = 1 ) & ( x <> 1 & y <> 1 implies f . (x,y) = min (x,y) ) )
hereby :: thesis: ( x <> 1 & y <> 1 implies f . (x,y) = min (x,y) )
assume A2: ( x = 1 or y = 1 ) ; :: thesis: f . (x,y) = 1
f . (x,y) = OpEx2 (x,y) by A1;
hence f . (x,y) = 1 by A2, OpEx2Def; :: thesis: verum
end;
assume A2: ( x <> 1 & y <> 1 ) ; :: thesis: f . (x,y) = min (x,y)
f . (x,y) = OpEx2 (x,y) by A1;
hence f . (x,y) = min (x,y) by OpEx2Def, A2; :: thesis: verum