
theorem
for X be non empty set, g be PartFunc of X,REAL, A be Subset of X holds
 |. g|A .| = (|.g.|)|A
proof
    let X be non empty set, g be PartFunc of X,REAL, A be Subset of X;

    reconsider gA = R_EAL(g|A) as PartFunc of X,ExtREAL;
A1: |.g|A.| = |. R_EAL(g|A) .| by MESFUNC5:def 7;
    dom(|.g|A.|) = dom (g|A) by MESFUNC1:def 10; then
A2: dom(|.g|A.|) = dom g /\ A by RELAT_1:61;
    dom((|.g.|)|A) = dom |.g.| /\ A by RELAT_1:61; then
A3: dom(|.g|A.|) = dom((|.g.|)|A) by A2,VALUED_1:def 11;
    for x be object st x in dom(|.g|A.|)
      holds (|.g|A.|).x = ((|.g.|)|A).x
    proof
     let x be object;
     assume
A4:  x in dom(|.g|A.|); then
     reconsider x1=x as Element of X;

     x in dom(g|A) by A4,MESFUNC1:def 10; then
A5:  x in dom g & x in A by RELAT_1:57; then
     x in dom (R_EAL g) by MESFUNC5:def 7; then
A6:  x in dom |. R_EAL g.| by MESFUNC1:def 10;
A7:  (|.g|A.|).x = |. (R_EAL(g|A)).x1 .| by A4,A1,MESFUNC1:def 10;
     (R_EAL(g|A)).x1 = (g|A).x1 by MESFUNC5:def 7; then
     (R_EAL(g|A)).x1 = g.x1 by A5,FUNCT_1:49; then
     |. (R_EAL(g|A)).x1 .| = |. (R_EAL g).x1 .| by MESFUNC5:def 7; then
     (|.g|A.|).x = (|. R_EAL g .|).x1 by A7,A6,MESFUNC1:def 10; then
     (|.g|A.|).x = (R_EAL abs g).x1 by MESFUNC6:1; then
     (|.g|A.|).x = (|.g.|).x1 by MESFUNC5:def 7;
     hence (|.g|A.|).x = ((|.g.|)|A).x by A5,FUNCT_1:49;
    end;
    hence |.g|A.| = (|.g.|)|A by A3,FUNCT_1:2;
end;
