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+;
reserve i,j,k for natural Ordinal;

theorem Th49:
  (i/j)*'(k/l) = (i*^k)/(j*^l)
proof
  per cases;
  suppose
A1: j <> {} & l <> {};
    then
A2: denominator (k/l) = RED(l,k) & denominator (i/j) = RED(j,i) by Th42;
    numerator (k/l) = RED(k,l) & numerator (i/j) = RED(i,j) by A1,Th42;
    hence
    (i/j)*'(k/l) = (RED(i,j)*^RED(k,l)*^(i hcf j))/(RED(j,i)*^RED(l,k)*^(
    i hcf j)) by A1,A2,Th15,Th44
      .= (RED(k,l)*^(RED(i,j)*^(i hcf j)))/(RED(j,i)*^RED(l,k)*^(i hcf j))
    by ORDINAL3:50
      .= (RED(k,l)*^i)/(RED(j,i)*^RED(l,k)*^(i hcf j)) by Th21
      .= (RED(k,l)*^i)/(RED(l,k)*^(RED(j,i)*^(i hcf j))) by ORDINAL3:50
      .= (RED(k,l)*^i)/(RED(l,k)*^j) by Th21
      .= (RED(k,l)*^i*^(l hcf k))/(RED(l,k)*^j*^(l hcf k)) by A1,Th15,Th44
      .= (i*^(RED(k,l)*^(l hcf k)))/(RED(l,k)*^j*^(l hcf k)) by ORDINAL3:50
      .= (i*^k)/(RED(l,k)*^j*^(l hcf k)) by Th21
      .= (i*^k)/(j*^(RED(l,k)*^(l hcf k))) by ORDINAL3:50
      .= (i*^k)/(j*^l) by Th21;
  end;
  suppose
A3: j = {} or l = {};
    then i/j = {} or k/l = {} by Def10;
    then
A4: (i/j)*'(k/l) = {} by Th48;
    j*^l = {} by A3,ORDINAL2:35;
    hence thesis by A4,Def10;
  end;
end;
