
theorem Th7:
for X be non empty set, x be Element of X, f1,f2 be Function of X,ExtREAL holds
  ( f1 is without-infty & f2 is without-infty implies (f1+f2).x = f1.x + f2.x )
& ( f1 is without+infty & f2 is without+infty implies (f1+f2).x = f1.x + f2.x )
& ( f1 is without-infty & f2 is without+infty implies (f1-f2).x = f1.x - f2.x )
& ( f1 is without+infty & f2 is without-infty implies (f1-f2).x = f1.x - f2.x )
proof
   let X be non empty set, x be Element of X,
       f1,f2 be Function of X,ExtREAL;
   hereby assume f1 is without-infty & f2 is without-infty; then
    reconsider F1=f1,F2=f2 as without-infty Function of X,ExtREAL;
    dom (F1+F2) = X by FUNCT_2:def 1;
    hence (f1+f2).x = f1.x + f2.x by MESFUNC1:def 3;
   end;
   hereby assume f1 is without+infty & f2 is without+infty; then
    reconsider F1=f1,F2=f2 as without+infty Function of X,ExtREAL;
    dom (F1+F2) = X by FUNCT_2:def 1;
    hence (f1+f2).x = f1.x + f2.x by MESFUNC1:def 3;
   end;
   hereby assume A1: f1 is without-infty & f2 is without+infty; then
    reconsider F1=f1 as without-infty Function of X,ExtREAL;
    reconsider F2=f2 as without+infty Function of X,ExtREAL by A1;
    dom (F1-F2) = X by FUNCT_2:def 1;
    hence (f1-f2).x = f1.x - f2.x by MESFUNC1:def 4;
   end;
   hereby assume A1: f1 is without+infty & f2 is without-infty; then
    reconsider F1=f1 as without+infty Function of X,ExtREAL;
    reconsider F2=f2 as without-infty Function of X,ExtREAL by A1;
    dom (F1-F2) = X by FUNCT_2:def 1;
    hence (f1-f2).x = f1.x - f2.x by MESFUNC1:def 4;
   end;
end;
