
theorem
for X,Y be non empty set, f1 be without-infty Function of [:X,Y:],ExtREAL,
 f2 be without+infty Function of [:X,Y:],ExtREAL
  holds ~(f1-f2) = ~f1 - ~f2 & ~(f2-f1) = ~f2 - ~f1
proof
   let X,Y be non empty set,
       f1 be without-infty Function of [:X,Y:],ExtREAL,
       f2 be without+infty Function of [:X,Y:],ExtREAL;
   now let z be Element of [:Y,X:];
    consider y,x be object such that
A1:  y in Y & x in X & z = [y,x] by ZFMISC_1:def 2;
    reconsider y as Element of Y by A1;
    reconsider x as Element of X by A1;
    reconsider z1 = [x,y] as Element of [:X,Y:] by ZFMISC_1:87;
    ~(f1-f2).z = ~(f1-f2).(y,x) by A1; then
    ~(f1-f2).z = (f1-f2).(x,y) by FUNCT_4:def 8; then
A2: ~(f1-f2).z = f1.z1 - f2.z1 by Th7;
    f1.z1 = f1.(x,y) & f2.z1 = f2.(x,y); then
    f1.z1 = ~f1.(y,x) & f2.z1 = ~f2.(y,x) by FUNCT_4:def 8;
    hence ~(f1-f2).z =(~f1 - ~f2).z by A1,A2,Th7;
   end;
   hence ~(f1-f2) = ~f1 - ~f2 by FUNCT_2:def 8;
   now let z be Element of [:Y,X:];
    consider y,x be object such that
A1:  y in Y & x in X & z = [y,x] by ZFMISC_1:def 2;
    reconsider y as Element of Y by A1;
    reconsider x as Element of X by A1;
    reconsider z1 = [x,y] as Element of [:X,Y:] by ZFMISC_1:87;
    ~(f2-f1).z = ~(f2-f1).(y,x) by A1; then
    ~(f2-f1).z = (f2-f1).(x,y) by FUNCT_4:def 8; then
A2: ~(f2-f1).z = f2.z1 - f1.z1 by Th7;
    f1.z1 = f1.(x,y) & f2.z1 = f2.(x,y); then
    f1.z1 = ~f1.(y,x) & f2.z1 = ~f2.(y,x) by FUNCT_4:def 8;
    hence ~(f2-f1).z =(~f2 - ~f1).z by A1,A2,Th7;
   end;
   hence ~(f2-f1) = ~f2 - ~f1 by FUNCT_2:def 8;
end;
