reserve r,p,x for Real;
reserve n for Element of NAT;
reserve A for non empty closed_interval Subset of REAL;
reserve Z for open Subset of REAL;

theorem Th20:
  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_1:12
    .= (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_1:12;
  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_1:12;
    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_1:13;
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_1:13
      .= 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;
