reserve x,y,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for PartFunc of C,COMPLEX;
reserve r1,r2,p1 for Real;
reserve r,q,cr1,cr2 for Complex;

theorem
  f1/f + f2/f = (f1 + f2)/f & f1/f - f2/f = (f1 - f2)/f
proof
  thus f1/f + f2/f = f1(#)(f^) +f2/f by Th38
    .= f1(#)(f^) + f2(#)(f^) by Th38
    .= (f1+f2) (#) (f^) by Th15
    .= (f1+f2)/f by Th38;
  thus f1/f - f2/f = f1(#)(f^) - f2/f by Th38
    .= f1(#)(f^) -f2(#)(f^) by Th38
    .= (f1-f2)(#)(f^) by Th19
    .= (f1-f2)/f by Th38;
end;
