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

theorem Th13:
  n <> {} implies (m*^n) div^ (m lcm n) divides m
proof
  assume
A1: n <> {};
  take ((m lcm n) div^ n);
  n divides m lcm n by Def4;
  then
A2: m lcm n = n*^((m lcm n) div^ n) by Th7;
  m lcm n divides m*^n by Th12;
  then m*^n = (m lcm n)*^ ((m*^n) div^ (m lcm n)) by Th7;
  then n*^m = n*^(((m lcm n) div^ n)*^ ((m*^n) div^ (m lcm n))) by A2,
ORDINAL3:50;
  hence m = ((m*^n) div^ (m lcm n))*^((m lcm n) div^ n) by A1,ORDINAL3:33;
end;
