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 Th45:
  j <> {} & l <> {} implies (i/j = k/l iff i*^l = j*^k)
proof
  assume that
A1: j <> {} and
A2: l <> {};
  set x = i/j, y = k/l;
  set ny = numerator y, dy = denominator y;
A3: ny = RED(k,l) by A2,Th42;
  set nx = numerator x, dx = denominator x;
A4: dx = RED(j,i) by A1,Th42;
A5: dy = RED(l,k) by A2,Th42;
A6: nx = RED(i,j) by A1,Th42;
  hereby
    assume i/j = k/l;
    then i = ny*^(i hcf j) & l = dx*^(l hcf k) by A6,A5,Th21;
    hence i*^l = ny*^(i hcf j)*^dx*^(l hcf k) by ORDINAL3:50
      .= ny*^((i hcf j)*^dx)*^(l hcf k) by ORDINAL3:50
      .= ny*^j*^(l hcf k) by A4,Th21
      .= j*^(ny*^(l hcf k)) by ORDINAL3:50
      .= j*^k by A3,Th21;
  end;
  assume
A7: i*^l = j*^k;
  then dx = RED(j*^l,j*^k) by A2,A4,Th28;
  then
A8: dx = dy by A1,A5,Th28;
  nx = RED(j*^k,j*^l) by A2,A6,A7,Th28;
  then nx = ny by A1,A3,Th28;
  then x = ny/dy by A8,Th39;
  hence thesis by Th39;
end;
