reserve F for Field,
  x for Element of F,
  V for VectSp of F,
  v for Element of V;

theorem
  for a, b, c being Element of F holds (a <> 0.F & a*c - b = 0.F implies
  c = b*a") & (a <> 0.F & b - c*a = 0.F implies c = b*a")
proof
  let a, b, c be Element of F;
  thus
A1: a <> 0.F & a*c - b = 0.F implies c = b*a"
  proof
    assume a <> 0.F;
    then
A2: a"*a = 1.F by Def10;
    assume a*c - b = 0.F;
    then a"*(a*c) = b*a" by RLVECT_1:21;
    then (a"*a)*c = b*a" by GROUP_1:def 3;
    hence thesis by A2;
  end;
  assume
A3: a <> 0.F;
  assume b - c*a = 0.F;
  then -(b - c*a) = 0.F by RLVECT_1:12;
  hence thesis by A1,A3,RLVECT_1:33;
end;
