reserve A,B,C for Ordinal;
reserve a,b,c,d for natural Ordinal;
reserve l,m,n for natural Ordinal;
reserve i,j,k for Element of omega;
reserve x,y,z for Element of RAT+;

theorem Th44:
  c <> {} implies (a*^c)/(b*^c) = a/b
proof
  assume
A1: c <> {};
  per cases;
  suppose
    b = {};
    then a/b = {} & b*^c = {} by Def10,ORDINAL2:35;
    hence thesis by Def10;
  end;
  suppose
A2: b <> {};
    then
A3: b*^c <> {} by A1,ORDINAL3:31;
    then
A4: denominator ((a*^c)/(b*^c)) = RED(b*^c,a*^c) by Th42
      .= RED(b,a) by A1,Th28
      .= denominator (a/b) by A2,Th42;
    numerator ((a*^c)/(b*^c)) = RED(a*^c,b*^c) by A3,Th42
      .= RED(a,b) by A1,Th28
      .= numerator (a/b) by A2,Th42;
    hence (a*^c)/(b*^c) = (numerator (a/b))/denominator (a/b) by A4,Th39
      .= a/b by Th39;
  end;
end;
