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

theorem Th102:
  i,j are_coprime & p|^n divides i*j implies p|^n divides i or p|^n divides j
  proof
    assume i,j are_coprime;
    then
A1: |.i.|,|.j.| are_coprime by INT_2:34;
    assume p|^n divides i*j;
    then p|^n divides |.i*j.| by Th4;
    then p|^n divides |.i.|*|.j.| by COMPLEX1:65;
    then p|^n divides |.i.| or p|^n divides |.j.| by A1,NUMBER08:107;
    hence thesis by Th4;
  end;
