
theorem Th33:
  for X,Y,Z be non empty set, F be Function of [:X,Y:],Z, y be Element of Y
   holds ProjMap2(F,y) = ProjMap1(~F,y)
proof
  let X,Y,Z be non empty set;
  let F be Function of [:X,Y:],Z;
  let y be Element of Y;
  now let x be Element of X;
   ProjMap2(F,y).x = F.(x,y) by MESFUNC9:def 7
    .= ~F.(y,x) by FUNCT_4:def 8;
   hence ProjMap2(F,y).x = ProjMap1(~F,y).x by MESFUNC9:def 6;
  end;
  hence ProjMap2(F,y) = ProjMap1(~F,y) by FUNCT_2:def 8;
end;
