
theorem Th4:
  for F being Field holds for a,b,c,d being Element
  of F holds a-b = c-d iff a+d = b+c
proof
  let F be Field;
  let a,b,c,d be Element of F;
  hereby
    assume a-b = c-d;
    then c-d+b = a+-b+b .= a+(b-b) by RLVECT_1:def 3
      .= a+0.F by RLVECT_1:5
      .= a by RLVECT_1:4;
    hence a+d = c+b+-d+d by RLVECT_1:def 3
      .= c+b+(d-d) by RLVECT_1:def 3
      .= c+b+0.F by RLVECT_1:5
      .= b+c by RLVECT_1:4;
  end;
  assume a+d = b+c;
  then b+c-d = a+(d-d) by RLVECT_1:def 3
    .= a+0.F by RLVECT_1:5
    .= a by RLVECT_1:4;
  hence a-b = c-d+b-b by RLVECT_1:def 3
    .= c-d+(b-b) by RLVECT_1:def 3
    .= c-d+0.F by RLVECT_1:5
    .= c-d by RLVECT_1:4;
end;
