reserve p,q,r for FinSequence;
reserve u,v,x,y,y1,y2,z for object, A,D,X,Y for set;
reserve i,j,k,l,m,n for Nat;

theorem Th57:
  len(p - A) = len p - card(p " A)
proof
  set q = Sgm(Seg(len p) \ p " A);
A1: Seg len p = dom p by FINSEQ_1:def 3;
    Seg(len p) \ p " A c= Seg(len p) by XBOOLE_1:36;
  then rng q c= dom p by A1,FINSEQ_1:def 14;
  then
A2: dom q = dom(p - A) by A1,RELAT_1:27;
A3: dom q = Seg len q by FINSEQ_1:def 3;
A4: p " A c= Seg len p
  proof
    let x be object;
A5: p " A c= dom p by RELAT_1:132;
A6: dom p = Seg len p by FINSEQ_1:def 3;
    assume x in p " A;
    hence thesis by A5,A6;
  end;
  len q = card(Seg(len p) \ p " A) by Th37;
  then len(p - A) = card(Seg(len p) \ p " A) by A2,A3,FINSEQ_1:def 3;
  hence len(p - A) = card(Seg(len p)) - card(p " A) by A4,CARD_2:44
    .= len p - card(p " A) by FINSEQ_1:57;
end;
