reserve A,B,C for Ordinal;
reserve a,b,c,d for natural Ordinal;
reserve l,m,n for natural Ordinal;

theorem Th20:
  a,b are_coprime iff a hcf b = 1
proof
  a hcf b divides b by Def5;
  then
A1: b = (a hcf b)*^(b div^ (a hcf b)) by Th7;
  a hcf b divides a by Def5;
  then a = (a hcf b)*^(a div^ (a hcf b)) by Th7;
  hence a,b are_coprime implies a hcf b = 1 by A1;
  assume
A2: a hcf b = 1;
  let c,d1,d2 be Ordinal such that
A3: a = c*^d1 & b = c*^d2;
  a <> {} or b <> {} by A2,Th14,Lm4;
  then reconsider c as Element of omega by A3,ORDINAL3:75;
  c divides a & c divides b by A3;
  then c divides 1 by A2,Def5;
  then ex d st 1 = c*^d by Th5;
  hence thesis by ORDINAL3:37;
end;
