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

theorem
  a <> {} or b <> {} implies ex c,d1,d2 being natural Ordinal st d1,d2
  are_coprime & a = c *^ d1 & b = c *^ d2
proof
  assume that
A1: a <> {} or b <> {} and
A2: not ex c,d1,d2 being natural Ordinal st d1,d2 are_coprime & a
  = c *^ d1 & b = c *^ d2;
A3: ex A st PP[A]
  proof
    per cases;
    suppose
A4:   a c= b;
      take A = b, B = a;
      thus B c= A & A in omega & A <> {} by A1,A4,ORDINAL1:def 12;
      thus thesis by A2;
    end;
    suppose
A5:   b c= a;
      take A = a, B = b;
      thus B c= A & A in omega & A <> {} by A1,A5,ORDINAL1:def 12;
      thus thesis by A2;
    end;
  end;
  consider A such that
A6: PP[A] and
A7: for C st PP[C] holds A c= C from ORDINAL1:sch 1(A3);
  consider B such that
A8: B c= A and
A9: A in omega and
A10: A <> {} and
A11: not ex c,d1,d2 being natural Ordinal st d1,d2 are_coprime &
  A = c *^ d1 & B = c *^ d2 by A6;
  reconsider A,B as Element of omega by A8,A9,ORDINAL1:12;
  A = 1 *^ A & B = 1 *^ B by ORDINAL2:39;
  then not A,B are_coprime by A11;
  then consider c,d1,d2 being Ordinal such that
A12: A = c *^ d1 and
A13: B = c *^ d2 and
A14: c <> 1;
A15: c <> {} by A10,A12,ORDINAL2:35;
  then
A16: d1 c= A & d2 c= B by A12,A13,ORDINAL3:36;
A17: d1 <> {} by A10,A12,ORDINAL2:38;
  then c c= A by A12,ORDINAL3:36;
  then reconsider c,d1,d2 as Element of omega by A16,ORDINAL1:12;
  1 in c or c in 1 by A14,ORDINAL1:14;
  then 1*^d1 in A by A12,A17,A15,ORDINAL3:14,19;
  then
A18: d1 in A by ORDINAL2:39;
A19: now
    let c9,d19,d29 be natural Ordinal;
    assume that
A20: d19,d29 are_coprime and
A21: d1 = c9 *^ d19 & d2 = c9 *^ d29;
    A = c*^c9*^d19 & B = c*^c9*^d29 by A12,A13,A21,ORDINAL3:50;
    hence contradiction by A11,A20;
  end;
  A = d1*^c & B = d2*^c by A12,A13;
  then d2 c= d1 by A8,A15,ORDINAL3:35;
  hence contradiction by A7,A17,A19,A18,ORDINAL1:5;
end;
