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

theorem Th28:
  c <> {} implies RED(a*^c, b*^c) = RED(a, b)
proof
  assume
A1: c <> {};
  a <> {} implies a hcf b <> {} by Th18;
  then
A2: a <> {} implies (a hcf b)*^c <> {} by A1,ORDINAL3:31;
A3: RED({},b) = {} & {}*^((a hcf b)*^c) = {} by ORDINAL2:35,ORDINAL3:70;
A4: a hcf b divides a by Def5;
  thus RED(a*^c, b*^c) = (a*^c)div^((a hcf b)*^c) by Th17
    .= (((a div^(a hcf b))*^(a hcf b))*^c)div^((a hcf b)*^c) by A4,Th7
    .= (RED(a,b)*^((a hcf b)*^c))div^((a hcf b)*^c) by ORDINAL3:50
    .= RED(a, b) by A2,A3,ORDINAL3:68,70;
end;
