
theorem Th49:
  for F being non degenerated almost_left_invertible commutative
Ring for a,b,c,d being Element of F st b <> 0.F & d <> 0.F holds (a/b) + (c/d)
  = (a*d + c*b) / (b * d)
proof
  let F be non degenerated almost_left_invertible commutative Ring;
  let a,b,c,d be Element of F;
  assume that
A1: b <> 0.F and
A2: d <> 0.F;
  (a*d + c*b) / (b * d) = (a*d + c*b) * (b" * d") by A1,A2,GCD_1:49
    .= ((a*d + c*b) * b") * d" by GROUP_1:def 3
    .= (((a*d) * b") + ((c*b) * b")) * d" by VECTSP_1:def 3
    .= (((a*d) * b") + (c* (b*b"))) * d" by GROUP_1:def 3
    .= (((a*d) * b") + (c*1.F)) * d" by A1,VECTSP_1:def 10
    .= (((a*d) * b") + c) * d"
    .= ((a*d) * b") * d" + c * d" by VECTSP_1:def 3
    .= b" * ((a*d) * d") + c * d" by GROUP_1:def 3
    .= b" * (a * (d*d")) + c * d" by GROUP_1:def 3
    .= b" * (a * 1.F) + c * d" by A2,VECTSP_1:def 10
    .= b" * a + c * d";
  hence thesis;
end;
