reserve x,y,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for PartFunc of C,COMPLEX;
reserve r1,r2,p1 for Real;
reserve r,q,cr1,cr2 for Complex;

theorem Th1:
  dom (f1+f2) = dom f1 /\ dom f2 & for c st c in dom(f1+f2) holds
  (f1+f2)/.c = (f1/.c) + (f2/.c)
proof
  thus
A1: dom (f1+f2) = dom f1 /\ dom f2 by VALUED_1:def 1;
  now
    let c;
    assume
A2: c in dom (f1+f2);
    then c in dom f1 by A1,XBOOLE_0:def 4;
    then
A3: f1.c = f1/.c by PARTFUN1:def 6;
    c in dom f2 by A1,A2,XBOOLE_0:def 4;
    then
A4: f2.c = f2/.c by PARTFUN1:def 6;
    thus (f1+f2)/.c = (f1+f2).c by A2,PARTFUN1:def 6
      .= f1/.c+f2/.c by A2,A3,A4,VALUED_1:def 1;
  end;
  hence thesis;
end;
