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 Th46:
  j <> {} & l <> {} implies (i/j)+(k/l) = (i*^l+^j*^k)/(j*^l)
proof
  assume that
A1: j <> {} and
A2: l <> {};
A3: denominator (k/l) = RED(l,k) & denominator (i/j) = RED(j,i) by A1,A2,Th42;
  numerator (k/l) = RED(k,l) & numerator (i/j) = RED(i,j) by A1,A2,Th42;
  hence
  (i/j)+(k/l) = (RED(i,j)*^RED(l,k)+^RED(j,i)*^RED(k,l))*^(i hcf j) /(RED
  (j,i)*^RED(l,k)*^(i hcf j)) by A1,A3,Th15,Th44
    .= (RED(i,j)*^RED(l,k)+^RED(j,i)*^RED(k,l))*^(i hcf j) /(RED(l,k)*^(RED(
  j,i)*^(i hcf j))) by ORDINAL3:50
    .= (RED(i,j)*^RED(l,k)+^RED(j,i)*^RED(k,l))*^(i hcf j)/(RED(l,k)*^j) by
Th21
    .= (RED(i,j)*^RED(l,k)*^(i hcf j)+^RED(j,i)*^RED(k,l)*^(i hcf j)) /(RED(
  l,k)*^j) by ORDINAL3:46
    .= (RED(l,k)*^(RED(i,j)*^(i hcf j))+^RED(j,i)*^RED(k,l)*^(i hcf j)) /(
  RED(l,k)*^j) by ORDINAL3:50
    .= (RED(l,k)*^i+^RED(j,i)*^RED(k,l)*^(i hcf j))/(RED(l,k)*^j) by Th21
    .= (RED(l,k)*^i+^RED(k,l)*^(RED(j,i)*^(i hcf j)))/(RED(l,k)*^j) by
ORDINAL3:50
    .= (RED(l,k)*^i+^RED(k,l)*^j)/(RED(l,k)*^j) by Th21
    .= ((RED(l,k)*^i+^RED(k,l)*^j)*^(k hcf l))/(RED(l,k)*^j*^(k hcf l)) by A2
,Th15,Th44
    .= ((RED(l,k)*^i+^RED(k,l)*^j)*^(k hcf l))/(j*^(RED(l,k)*^(k hcf l))) by
ORDINAL3:50
    .= ((RED(l,k)*^i+^RED(k,l)*^j)*^(k hcf l))/(j*^l) by Th21
    .= (RED(l,k)*^i*^(k hcf l)+^RED(k,l)*^j*^(k hcf l))/(j*^l) by ORDINAL3:46
    .= (i*^(RED(l,k)*^(k hcf l))+^RED(k,l)*^j*^(k hcf l))/(j*^l) by ORDINAL3:50
    .= (i*^l+^RED(k,l)*^j*^(k hcf l))/(j*^l) by Th21
    .= (i*^l+^j*^(RED(k,l)*^(k hcf l)))/(j*^l) by ORDINAL3:50
    .= (i*^l+^j*^k)/(j*^l) by Th21;
end;
