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