
theorem Th8:
for X be non empty set, f1,f2 be without-infty Function of X,ExtREAL
 holds f1 + f2 = f1 - (-f2) & -(f1 + f2) = -f1 - f2
proof
   let X be non empty set, f1,f2 be without-infty Function of X,ExtREAL;
   thus f1 + f2 = f1 - (-f2) by Lm3;
A1:dom(-f1) = X by FUNCT_2:def 1;
A2:dom(-f2) = X by FUNCT_2:def 1;
A3:dom(-(f1+f2)) = X by FUNCT_2:def 1;
   now let x be Element of X;
    (-(f1+f2)).x = -((f1+f2).x) by A3,MESFUNC1:def 7
     .= -(f1.x + f2.x) by Th7
     .= -(f1.x) - (f2.x) by XXREAL_3:25
     .= (-f1).x + -(f2.x) by A1,MESFUNC1:def 7
     .= (-f1).x + (-f2).x by A2,MESFUNC1:def 7
     .= ((-f1) + (-f2)).x by Th7;
    hence (-(f1+f2)).x = (-f1-f2).x by Lm5;
   end;
   hence thesis by FUNCT_2:def 8;
end;
