reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve p for Prime;

theorem Th17:
  m <= n implies (primesFinS n) | m = primesFinS m
  proof
    set N = primesFinS n;
    set M = primesFinS m;
    assume
A1: m <= n;
A2: len N = n by Def1;
A3: len M = m by Def1;
    hence
A4: len (N|m) = len M by A1,A2,FINSEQ_1:59;
    let k such that
A5: 1 <= k and
A6: k <= len(N|m);
    reconsider z = k-1 as Element of NAT by A5,INT_1:3;
A7: k in Seg m by A3,A4,A5,A6;
A8: z < m-0 by A3,A4,A6,XREAL_1:8;
    then
A9: z < n by A1,XXREAL_0:2;
A10: k = z+1;
    thus (N|m).k = N.k by A7,FUNCT_1:49
    .= primenumber z by A9,A10,Def1
    .= M.k by A8,A10,Def1;
  end;
