reserve a,b,r for non unit non zero Real;
reserve X for non empty set,
        x for Tuple of 4,X;
reserve V             for RealLinearSpace,
        A,B,C,P,Q,R,S for Element of V;

theorem Th11:
  P,Q,R are_collinear & P <> R implies
  affine-ratio(R,Q,P) = 1 - affine-ratio(P,Q,R)
  proof
    assume that
A1: P,Q,R are_collinear and
A2: P <> R;
    set r = affine-ratio(P,Q,R),
        s = affine-ratio(R,Q,P);
A3: Q - P = r * (R - P) by A1,A2,Def02;
A4: Q - P = Q + 0.V - P
         .= Q + (-R + R) - P by RLVECT_1:5
         .= Q + (-R) + R - P by RLVECT_1:def 3
         .= (Q - R) + (R - P) by RLVECT_1:def 3
         .= (Q - R) - (P - R) by RLVECT_1:33;
    R,Q,P are_collinear by A1;
    then r * (R - P) - s * (P - R) = s * (P - R) - (P - R) - s * (P - R)
                                    by A3,A4,A2,Def02
                                  .= s * (P - R) + (R - P) - s * (P - R)
                                    by RLVECT_1:33
                                  .= s * (P - R) + ((R - P) - s * (P - R))
                                    by RLVECT_1:def 3
                                  .= s * (P - R) + (- s * (P - R) + (R - P))
                                    by RLVECT_1:def 2
                                  .= s * (P - R) - s * (P - R) + (R - P)
                                    by RLVECT_1:def 3
                                  .= 0.V + (R - P) by RLVECT_1:5
                                  .= R - P;
    then R - P = r * (R - P) - s * (-(R - P)) by RLVECT_1:33
              .= r * (R - P) - (s * ((-1) * (R - P))) by RLVECT_1:16
              .= r * (R - P) - (s * (-1)) * (R - P) by RLVECT_1:def 7
              .= r * (R - P) - (-s) * (R - P);
    then (R - P) + (-s) * (R - P)
      = r * (R - P) + (- (-s) * (R - P) + (-s) * (R - P)) by RLVECT_1:def 3
     .= r * (R - P) + 0.V by RLVECT_1:5
     .= r * (R - P);
    then 1 * (R - P) +  (-s) * (R - P) = r * (R - P) by RLVECT_1:def 8;
    then (1 - s) * (R - P) = r * (R - P) by RLVECT_1:def 6;
    then 1 - s = r by A2,Th08;
    hence thesis;
  end;
