
theorem Th32:
  for X,Y,Z be non empty set, F be Function of [:X,Y:],Z, x be Element of X
   holds ProjMap1(F,x) = ProjMap2(~F,x)
proof
  let X,Y,Z be non empty set;
  let F be Function of [:X,Y:],Z;
  let x be Element of X;
  now let y be Element of Y;
   ProjMap1(F,x).y = F.(x,y) by MESFUNC9:def 6
    .= ~F.(y,x) by FUNCT_4:def 8;
   hence ProjMap1(F,x).y = ProjMap2(~F,x).y by MESFUNC9:def 7;
  end;
  hence ProjMap1(F,x) = ProjMap2(~F,x) by FUNCT_2:def 8;
end;
