
theorem Th12:
  for S be non empty finite set,
  s be FinSequence of S,
  A be Subset of (dom s),
  f be Function st f=canFS(A)
  holds extract(s,A)*f" = s|A
  proof
    let S be non empty finite set,
    s be FinSequence of S,
    A be Subset of (dom s),
    f be Function;
    assume A1: f=canFS(A);
    A2:(f)*(f") =id (rng f) by A1,FUNCT_1:39
    .= id A by A1,FUNCT_2:def 3;
    A3: dom (s|A) = (dom s) /\ A by RELAT_1:61
    .= dom(s*(id A)) by FUNCT_1:19;
    now let x be object;
      assume A4: x in dom (s|A);
      then A5: x in (dom s) /\ A by RELAT_1:61;
      thus (s|A).x = s.x by A4,FUNCT_1:47
      .=(s*(id A)).x by A5,FUNCT_1:20;
    end; then
    A6: s*(id A) = s|A by A3,FUNCT_1:2;
    thus extract(s,A)*f" = s|A by A6,A2,A1,RELAT_1:36;
  end;
