
theorem
for X,Y be non empty set, f1,f2 be PartFunc of X,ExtREAL st
  dom f1 c= Y & f2 = Y --> 0 holds f1+f2 = f1 & f1-f2 = f1 & f2-f1 = -f1
proof
    let X,Y be non empty set, f1,f2 be PartFunc of X,ExtREAL;
    assume that
A1:  dom f1 c= Y and
A2:  f2 = Y --> 0;
A3: dom f2 = Y by A2,FUNCOP_1:13;
    f2 is without-infty without+infty by A2,Th21; then
A4: dom(f1+f2) = dom f1 /\ dom f2
  & dom(f1-f2) = dom f1 /\ dom f2
  & dom(f2-f1) = dom f1 /\ dom f2 by Th23; then
A5: dom(f1+f2) = dom f1 & dom(f1-f2) = dom f1 & dom(f2-f1) = dom f1
      by A1,A3,XBOOLE_1:28; then
A6: dom(-f1) = dom(f2-f1) by MESFUNC1:def 7;
    now let x be Element of X;
     assume A7: x in dom(f1+f2); then
A8:  f2.x = 0 by A1,A2,A5,FUNCOP_1:7;
     (f1+f2).x = f1.x + f2.x by A7,MESFUNC1:def 3;
     hence (f1+f2).x = f1.x by A8,XXREAL_3:4;
    end;
    hence f1+f2 = f1 by A4,A1,A3,XBOOLE_1:28,PARTFUN1:5;
    now let x be Element of X;
     assume A9: x in dom(f1-f2); then
A10: f2.x = 0 by A1,A2,A5,FUNCOP_1:7;
     (f1-f2).x = f1.x - f2.x by A9,MESFUNC1:def 4;
     hence (f1-f2).x = f1.x by A10,XXREAL_3:15;
    end;
    hence f1-f2 = f1 by A4,A1,A3,XBOOLE_1:28,PARTFUN1:5;
    now let x be Element of X;
     assume A11: x in dom(f2-f1); then
A12: f2.x = 0 by A1,A2,A5,FUNCOP_1:7;
     (f2-f1).x = f2.x - f1.x by A11,MESFUNC1:def 4
      .= -f1.x + f2.x by XXREAL_3:def 4
      .= -f1.x by A12,XXREAL_3:4;
     hence (f2-f1).x = (-f1).x by A11,A6,MESFUNC1:def 7;
    end;
    hence f2-f1 = -f1 by A6,PARTFUN1:5;
end;
