reserve A,B for Ordinal,
  K,M,N for Cardinal,
  x,x1,x2,y,y1,y2,z,u for object,X,Y,Z,X1,X2, Y1,Y2 for set,
  f,g for Function;
reserve m,n for Nat;

theorem Th43:
  for X,Y being finite set st Y c= X holds card (X \ Y) = card X - card Y
proof
  let X,Y be finite set;
  defpred P[set] means ex S being finite set st S = $1 & card (X \ S) = card X
  - card S;
  card X - 0 = card X & X \ {} = X;
  then
A1: P[{}] by CARD_1:27;
  assume
A2: Y c= X;
A3: for X1,Z being set st X1 in Y & Z c= Y & P[Z] holds P[Z \/ {X1}]
  proof
    let X1,Z be set such that
A4: X1 in Y and
A5: Z c= Y and
A6: P[Z] and
A7: not P[Z \/ {X1}];
A8: card {X1} = 1 by CARD_1:30;
A9: now
      assume X1 in Z;
      then {X1} c= Z by ZFMISC_1:31;
      then Z = Z \/ {X1} by XBOOLE_1:12;
      hence P[Z \/ {X1}] by A6;
    end;
    then
A10: X1 in X \ Z by A2,A4,A7,XBOOLE_0:def 5;
    then consider m being Nat such that
A11: card (X \ Z) = m+1 by NAT_1:6;
    reconsider Z1 = Z as finite set by A5;
A12: X \ Z, card (X \ Z) are_equipotent & X \ (Z \/ {X1}) = X \ Z \ {X1}
    by CARD_1:def 2,XBOOLE_1:41;
    card { X1} = 1 by CARD_1:30;
    then
A13: card Z1 + card {X1} = card (Z1 \/ {X1}) by A7,A9,Th40;
    Segm(m+1) = succ Segm m by NAT_1:38;
    then m in m+1 & m = m+1 \ {m} by ORDINAL1:6,37;
    then X \ (Z \/ {X1}), m are_equipotent by A10,A11,A12,CARD_1:34;
    then card (X \ (Z \/ {X1})) = card X - card (Z1 \/ {X1}) by A6,A13,A11,A8,
CARD_1:def 2;
    hence contradiction by A7;
  end;
A14: Y is finite;
  P[Y] from FINSET_1:sch 2(A14,A1,A3);
  hence thesis;
end;
