
theorem Th34:
for X,Y be non empty set, F be Function of [:X,Y:],ExtREAL, x be Element of X
 holds ProjMap1(-F,x) = -ProjMap1(F,x)
proof
   let X,Y be non empty set, F be Function of [:X,Y:],ExtREAL,
       x be Element of X;
   now let y be Element of Y;
    [x,y] in [:X,Y:] by ZFMISC_1:87; then
A1: [x,y] in dom(-F) by FUNCT_2:def 1;
A2: dom(-ProjMap1(F,x)) = Y by FUNCT_2:def 1;
    ProjMap1(-F,x).y = (-F).(x,y) by MESFUNC9:def 6; then
    ProjMap1(-F,x).y = -(F.(x,y)) by A1,MESFUNC1:def 7; then
    ProjMap1(-F,x).y = -(ProjMap1(F,x).y) by MESFUNC9:def 6;
    hence ProjMap1(-F,x).y = (-ProjMap1(F,x)).y by A2,MESFUNC1:def 7;
   end;
   hence ProjMap1(-F,x) = -ProjMap1(F,x) by FUNCT_2:def 8;
end;
