let f1, f2 be BinOp of {0,1,2}; :: thesis: ( ( for x, y being Element of {0,1,2} holds
( ( ( x = 1 or y = 1 ) implies f1 . (x,y) = 1 ) & ( x <> 1 & y <> 1 implies f1 . (x,y) = min (x,y) ) ) ) & ( for x, y being Element of {0,1,2} holds
( ( ( x = 1 or y = 1 ) implies f2 . (x,y) = 1 ) & ( x <> 1 & y <> 1 implies f2 . (x,y) = min (x,y) ) ) ) implies f1 = f2 )
assume that
A1: for x, y being Element of {0,1,2} holds
( ( ( x = 1 or y = 1 ) implies f1 . (x,y) = 1 ) & ( x <> 1 & y <> 1 implies f1 . (x,y) = min (x,y) ) ) and
A2: for x, y being Element of {0,1,2} holds
( ( ( x = 1 or y = 1 ) implies f2 . (x,y) = 1 ) & ( x <> 1 & y <> 1 implies f2 . (x,y) = min (x,y) ) ) ; :: thesis: f1 = f2
for x, y being Element of {0,1,2} holds f1 . (x,y) = f2 . (x,y)

proof

let x, y be Element of {0,1,2}; :: thesis: f1 . (x,y) = f2 . (x,y)

per cases ( x = 1 or y = 1 or ( x <> 1 & y <> 1 ) ) ;

suppose A3: ( x = 1 or y = 1 ) ; :: thesis: f1 . (x,y) = f2 . (x,y)

then f1 . (x,y) = 1 by A1
.= f2 . (x,y) by A2, A3 ;
hence f1 . (x,y) = f2 . (x,y) ; :: thesis: verum

end;

suppose A3: ( x <> 1 & y <> 1 ) ; :: thesis: f1 . (x,y) = f2 . (x,y)

then f1 . (x,y) = min (x,y) by A1
.= f2 . (x,y) by A2, A3 ;
hence f1 . (x,y) = f2 . (x,y) ; :: thesis: verum

end;

end;

end;

hence f1 = f2 by BINOP_1:2; :: thesis: verum