
theorem Th6:
for X be set, f1,f2 be PartFunc of X,REAL st dom f1 = dom f2 holds
 (f1 + f2) - f2 = f1 & (f1 - f2) + f2 = f1
proof
    let X be set, f1,f2 be PartFunc of X,REAL;
    assume A1: dom f1 = dom f2;
A2: dom(f1+f2) = dom f1 /\ dom f1 by A1,VALUED_1:def 1 .= dom f1; then
A3: dom((f1+f2)-f2) = dom f1 /\ dom f1 by A1,VALUED_1:12 .= dom f1;
A4: dom(f1-f2) = dom f1 /\ dom f1 by A1,VALUED_1:12 .= dom f1; then
A5: dom((f1-f2)+f2) = dom f1 /\ dom f1 by A1,VALUED_1:def 1 .= dom f1;

    for x be Element of X st x in dom f1 holds f1.x = ((f1+f2)-f2).x
    proof
     let x be Element of X;
     assume A6: x in dom f1; then
     ((f1+f2)-f2).x = (f1+f2).x - f2.x by A3,VALUED_1:13
      .= f1.x + f2.x - f2.x by A6,A2,VALUED_1:def 1;
     hence f1.x = ((f1+f2)-f2).x;
    end;
    hence (f1 + f2) - f2 = f1 by A3,PARTFUN1:5;

    for x be Element of X st x in dom f1 holds f1.x = ((f1-f2)+f2).x
    proof
     let x be Element of X;
     assume A7: x in dom f1; then
     ((f1-f2)+f2).x = (f1-f2).x + f2.x by A5,VALUED_1:def 1
      .= f1.x - f2.x + f2.x by A4,A7,VALUED_1:13;
     hence f1.x = ((f1-f2)+f2).x;
    end;
    hence (f1 - f2) + f2 = f1 by A5,PARTFUN1:5;
end;
