reserve k, l, m, n, i, j for Nat,
  K, N for non empty Subset of NAT,
  Ke, Ne, Me for Subset of NAT,
  X,Y for set;
reserve f for Function of Segm n,Segm k;
reserve x,y for set;

theorem Th55:
  card X=k+1 & x in X implies card (X\{x})=k
proof
  assume that
A1: card X=k+1 and
A2: x in X;
  reconsider X9=X as finite set by A1;
  set Xx=X9\{x};
  {x} c= X by A2,ZFMISC_1:31;
  then {x} /\X ={x} by XBOOLE_1:28;
  then Xx \/ {x} = X by XBOOLE_1:51;
  then
A3: card {x}+ card Xx= k+1 by A1,CARD_2:40,XBOOLE_1:79;
  card {x}=1 by CARD_1:30;
  hence thesis by A3;
end;
