
theorem
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
 E be Element of S, f,g be PartFunc of X,REAL
  st (E = dom f or E = dom g) & f a.e.= g,M
  holds (f-g) a.e.= (X-->0)|E,M
proof
    let X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
    E be Element of S, f,g be PartFunc of X,REAL;
    assume that
A1:  E = dom f or E = dom g and
A2:  f a.e.= g,M;
    consider A be Element of S such that
A3:  M.A = 0 & f|A` = g|A` by A2,LPSPACE1:def 10;

A4: dom f /\ A` = dom(g|A`) by A3,RELAT_1:61
     .= dom g /\ A` by RELAT_1:61;

A5: dom((f-g)|A`) = dom(f-g) /\ A` by RELAT_1:61; then
A6: dom((f-g)|A`)
      = (dom f /\ dom g) /\ A` by VALUED_1:12
     .= (dom f /\ A`) /\ (dom f /\ A`) by A4,XBOOLE_1:116;

A7: dom((f-g)|A`)
      = (dom f /\ dom g) /\ A` by A5,VALUED_1:12
     .= (dom g /\ A`) /\ (dom g /\ A`) by A4,XBOOLE_1:116;

A8: dom(((X-->0)|E)|A`) = dom((X-->0)|E) /\ A` by RELAT_1:61
     .= dom(X-->0) /\ E /\ A` by RELAT_1:61
     .= E /\ A` by XBOOLE_1:28;

    for x be Element of X st x in dom((f-g)|A`) holds
     ((f-g)|A`).x = (((X-->0)|E)|A`).x
    proof
     let x be Element of X;
     assume A9: x in dom((f-g)|A`); then
A10: x in E by A7,A1,A6,XBOOLE_0:def 4;
A11:  x in dom(f-g) & x in A` by A5,A9,XBOOLE_0:def 4; then
A12:  ((f-g)|A`).x = (f-g).x by FUNCT_1:49
      .= f.x - g.x by A11,VALUED_1:13
      .= (f|A`).x - g.x by A11,FUNCT_1:49
      .= (f|A`).x - (g|A`).x by A11,FUNCT_1:49
      .= 0 by A3;
     (((X-->0)|E)|A`).x = ((X-->0)|E).x by A11,FUNCT_1:49
      .= (X-->0).x by A10,FUNCT_1:49;
     hence ((f-g)|A`).x = (((X-->0)|E)|A`).x by A12;
    end;
    hence (f-g) a.e.= (X-->0)|E,M by A3,A7,A1,A6,A8,PARTFUN1:5,LPSPACE1:def 10;
end;
