
theorem Th35:
  for X,Y be non empty set, F be Function of [:X,Y:],ExtREAL, y be Element of Y
   holds ProjMap2(-F,y) = -ProjMap2(F,y)
proof
   let X,Y be non empty set, F be Function of [:X,Y:],ExtREAL,
       y be Element of Y;
   now let x be Element of X;
    [x,y] in [:X,Y:] by ZFMISC_1:87; then
A1: [x,y] in dom(-F) by FUNCT_2:def 1;
A2: dom(-ProjMap2(F,y)) = X by FUNCT_2:def 1;
    ProjMap2(-F,y).x = (-F).(x,y) by MESFUNC9:def 7; then
    ProjMap2(-F,y).x = -(F.(x,y)) by A1,MESFUNC1:def 7; then
    ProjMap2(-F,y).x = -(ProjMap2(F,y).x) by MESFUNC9:def 7;
    hence ProjMap2(-F,y).x = (-ProjMap2(F,y)).x by A2,MESFUNC1:def 7;
   end;
   hence ProjMap2(-F,y) = -ProjMap2(F,y) by FUNCT_2:def 8;
end;
