
theorem Cosik1:
  for n, m being Nat holds n < m iff primenumber n < primenumber m
  proof
    let n, m be Nat;
AA: for n, m being Nat st n < m holds primenumber n < primenumber m
    proof
      let n, m be Nat;
      assume
A2:   n < m;
      assume
A3:   primenumber n >= primenumber m;
      n = card SetPrimenumber primenumber n by NEWTON:def 8; then
A4:   card SetPrimenumber primenumber n < card SetPrimenumber primenumber m
        by A2,NEWTON:def 8;
      Segm card SetPrimenumber primenumber m c=
        Segm card SetPrimenumber primenumber n by CARD_1:11,A3,NEWTON:69;
      hence contradiction by A4,NAT_1:39;
    end;
    hence n < m implies primenumber n < primenumber m;
    assume
A1: primenumber n < primenumber m;
    assume m <= n; then
    m < n or m = n by XXREAL_0:1;
    hence thesis by A1,AA;
  end;
