reserve X for set;
reserve n,i for Element of NAT;
reserve a,b,c,d,e,r,x0 for Real;
reserve A for non empty closed_interval Subset of REAL;
reserve f,g,h for PartFunc of REAL,REAL n;
reserve E for Element of REAL n;

theorem Th18:
  A c= dom h implies |. h|A .| = (|.h.|) |A
  proof
    assume A1:  A c= dom h;
A2: dom (h|A) = A by A1,RELAT_1:62;
A3: dom ((|.h.|) |A) = dom (|.h.|) /\ A by RELAT_1:61
    .= dom (h) /\ A by NFCONT_4:def 2
    .= A by A1,XBOOLE_1:28;
A4: dom |. h|A .| = dom ((|.h.|) |A) by A3,A2,NFCONT_4:def 2;
    now let x be object;
      assume A5: x in dom ((|.h.|) |A);
      then
A6:   x in dom (|.h.|) /\ A by RELAT_1:61;
      then
A7:   x in dom (h) /\ A by NFCONT_4:def 2;
      then
A8:   x in dom (h|A) by RELAT_1:61;
A9:   x in dom (|.h.|) by A6,XBOOLE_0:def 4;
A10:  x in dom h by A7,XBOOLE_0:def 4;
A11:  h/.x = h.x by A10,PARTFUN1:def 6
      .= (h|A).x by A8,FUNCT_1:47
      .= (h|A)/.x by A8,PARTFUN1:def 6;
      thus ((|.h.|) |A).x = (|.h.|).x by A6,FUNCT_1:48
      .= (|.h.|)/.x by A9,PARTFUN1:def 6
      .= |. h/.x .| by A9,NFCONT_4:def 2
      .= (|. (h|A) .|) /.x by A11,A4,A5,NFCONT_4:def 2
      .= (|. (h|A) .|) .x by A4,A5,PARTFUN1:def 6;
    end;
    hence thesis by A4,FUNCT_1:2;
  end;
