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

theorem
  for m being positive Nat, k being Nat
  ex a,b being positive Nat st 2*k = a-b & a,m are_coprime & b,m are_coprime
  proof
    let m be positive Nat;
    let k be Nat;
    set S = the Sierp49FS of m,k;
    set q = PrimeDivisorsFS(m);
    per cases;
    suppose q is non empty;
      then reconsider q as CR_Sequence;
      ex a,b being positive Nat st
      2*k = a-b & a,m are_coprime & b,m are_coprime &
      a = CRT(S,q) + Product(q) + 2*k & b = CRT(S,q) + Product(q) by Th54;
      hence thesis;
    end;
    suppose q is empty;
      then
A1:   m = 1 by Th50;
      take a = 2*k+2;
      take b = 2;
      thus 2*k = a-b;
      thus thesis by A1;
    end;
  end;
