
theorem Th3:
  for i,j being Nat, b being bag of j st i <= j holds (b
  |i) is Element of Bags i
proof
  let i,j being Nat, b being bag of j;
  assume
A1: i <= j;
  Segm i c= Segm j
  proof
    let x be object;
    assume x in Segm i;
    then x in {y where y is Nat : y < i } by AXIOMS:4;
    then consider x9 being Nat such that
A2: x9=x and
A3: x9 < i;
    x9 < j by A1,A3,XXREAL_0:2;
    then x9 in {y where y is Nat : y < j };
    hence thesis by A2,AXIOMS:4;
  end;
  hence thesis by PRE_POLY:def 12;
end;
