
theorem Th36:
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
 f,g be PartFunc of X,ExtREAL, E be Element of S
  st f|E = g|E & E c= dom f & E c= dom g & f is E-measurable
  holds g is E-measurable
proof
    let X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
     f,g be PartFunc of X,ExtREAL, A be Element of S;
    assume that
A1:  f|A = g|A and
A2:  A c= dom f and
A3:  A c= dom g and
A4:  f is A-measurable;

    now let r be Real;
     now let x be object;
      assume x in A /\ less_dom(f,r); then
A5:   x in A & x in less_dom(f,r) by XBOOLE_0:def 4; then
A6:   x in dom f & f.x < r by MESFUNC1:def 11;
      f.x = (f|A).x by A5,FUNCT_1:49; then
      f.x = g.x by A1,A5,FUNCT_1:49; then
      x in less_dom(g,r) by A3,A5,A6,MESFUNC1:def 11;
      hence x in A /\ less_dom(g,r) by A5,XBOOLE_0:def 4;
     end; then
A7:  A /\ less_dom(f,r) c= A /\ less_dom(g,r);

     now let x be object;
      assume x in A /\ less_dom(g,r); then
A8:   x in A & x in less_dom(g,r) by XBOOLE_0:def 4; then
A9:   x in dom g & g.x < r by MESFUNC1:def 11;
      g.x = (g|A).x by A8,FUNCT_1:49; then
      g.x = f.x by A1,A8,FUNCT_1:49; then
      x in less_dom(f,r) by A2,A8,A9,MESFUNC1:def 11;
      hence x in A /\ less_dom(f,r) by A8,XBOOLE_0:def 4;
     end; then
     A /\ less_dom(g,r) c= A /\ less_dom(f,r); then
     A /\ less_dom(g,r) = A /\ less_dom(f,r) by A7;
     hence A /\ less_dom(g,r) in S by A4,MESFUNC1:def 16;
    end;
    hence thesis by MESFUNC1:def 16;
end;
