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
  for X,Y being finite set st X c< Y holds card X < card Y &
   card X in Segm card Y
proof
  let X,Y be finite set;
  assume
A1: X c< Y;
  then X c= Y;
  then
A2: Y = X \/ (Y\X) by XBOOLE_1:45;
  then
A3: card Y = card X + card (Y\X) by Th39,XBOOLE_1:79;
  then
A4: card X <= card Y by NAT_1:11;
  now
    assume card (Y\X) = 0;
    then Y \ X = {};
    hence contradiction by A1,A2;
  end;
  then card X <> card Y by A3;
  hence card X < card Y by A4,XXREAL_0:1;
  hence thesis by NAT_1:44;
end;
