reserve x for object, X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for complex-valued Function;
reserve r,p for Complex;

theorem
  f1/f - g1/g = (f1(#)g - g1(#)f)/(f(#)g)
proof
  thus f1/f - g1/g = f1/f + ((-1)(#) g1)/g by Th32
    .= (f1(#)g + (-1)(#) g1(#)f)/(f(#)g) by Th40
    .= (f1(#)g - (g1(#)f))/(f(#)g) by Th12;
end;
