reserve i,k,n,m for Element of NAT;
reserve a,b,r,r1,r2,s,x,x1,x2 for Real;
reserve A for non empty closed_interval Subset of REAL;
reserve X for set;

theorem Th5:
  for f,g being PartFunc of REAL,REAL, C being non empty Subset of
  REAL holds (f+g)||C = f||C + g||C
proof
  let f,g be PartFunc of REAL,REAL;
  let C be non empty Subset of REAL;
A1: dom (f||C+g||C) =dom (f|C) /\ dom (g||C) by VALUED_1:def 1
    .= (dom f /\ C) /\ dom (g|C) by RELAT_1:61
    .= (dom f /\ C) /\ (dom g /\ C) by RELAT_1:61
    .= dom f /\ (C /\ (dom g /\ C)) by XBOOLE_1:16
    .= dom f /\ (dom g /\ (C /\ C)) by XBOOLE_1:16
    .= dom f /\ (dom g /\ C);
A2: dom ((f+g)||C) = dom (f+g) /\ C by RELAT_1:61
    .= dom f /\ dom g /\ C by VALUED_1:def 1;
  then
A3: dom ((f+g)||C) = dom (f||C + g||C) by A1,XBOOLE_1:16;
  for c being Element of C st c in dom ((f+g)||C) holds (f+g)||C.c = (f||C
  + g||C).c
  proof
    let c be Element of C;
    assume
A4: c in dom((f+g)||C);
    then c in dom(f+g) /\ C by RELAT_1:61;
    then
A5: c in dom(f+g) by XBOOLE_0:def 4;
A6: c in dom(f||C) /\ dom(g||C) by A3,A4,VALUED_1:def 1;
    then
A7: c in dom(f||C) by XBOOLE_0:def 4;
A8: (f+g)||C.c = (f+g).c by A4,FUNCT_1:47
      .= f.c + g.c by A5,VALUED_1:def 1;
A9: c in dom(g||C) by A6,XBOOLE_0:def 4;
    (f||C+g||C).c =f|C.c + g||C.c by A3,A4,VALUED_1:def 1
      .= f.c + g|C.c by A7,FUNCT_1:47;
    hence thesis by A8,A9,FUNCT_1:47;
  end;
  hence thesis by A2,A1,PARTFUN1:5,XBOOLE_1:16;
end;
