reserve x,y for set,
        r,s for Real,
        S for non empty addLoopStr,
        LS,LS1,LS2 for Linear_Combination of S,
        G for Abelian add-associative right_zeroed right_complementable
          non empty addLoopStr,
        LG,LG1,LG2 for Linear_Combination of G,
        g,h for Element of G,
        RLS for non empty RLSStruct,
        R for vector-distributive scalar-distributive scalar-associative
        scalar-unitalnon empty RLSStruct,
        AR for Subset of R,
        LR,LR1,LR2 for Linear_Combination of R,
        V for RealLinearSpace,
        v,v1,v2,w,p for VECTOR of V,
        A,B for Subset of V,
        F1,F2 for Subset-Family of V,
        L,L1,L2 for Linear_Combination of V;
reserve I for affinely-independent Subset of V;

theorem
  not w in Affin A & v1 in A & v2 in A & r<>1 & r*w + (1-r)*v1 = s*w + (1-s)*v2
    implies r = s & v1 = v2
 proof
  assume that
   A1: (not w in Affin A) & v1 in A & v2 in A and
   A2: r<>1 and
   A3: r*w+(1-r)*v1=s*w+(1-s)*v2;
  r*w=r*w+0.V
   .=r*w+((1-r)*v1-(1-r)*v1) by RLVECT_1:15
   .=(s*w+(1-s)*v2)-(1-r)*v1 by A3,RLVECT_1:28
   .=((1-s)*v2-(1-r)*v1)+s*w by RLVECT_1:28;
  then  r*w-s*w=(1-s)*v2-(1-r)*v1 by RLVECT_4:1;
  then A4: (r-s)*w=(1-s)*v2-(1-r)*v1 by RLVECT_1:35;
  A5: A c=Affin A by Lm7;
  per cases;
  suppose r<>s;
   then A6: r-s<>0;
   A7: 1/(r-s)*(1-s)=(r-s-(r-1))/(r-s) by XCMPLX_1:99
    .=(r-s)/(r-s)-(r-1)/(r-s) by XCMPLX_1:120
    .=1-(r-1)/(r-s) by A6,XCMPLX_1:60;
   A8: -(1/(r-s)*(1-r))=-((1-r)/(r-s)) by XCMPLX_1:99
    .=(-(1-r))/(r-s) by XCMPLX_1:187;
   1=(r-s)*(1/(r-s)) by A6,XCMPLX_1:106;
   then w=(1/(r-s)*(r-s))*w by RLVECT_1:def 8
    .=1/(r-s)*((r-s)*w) by RLVECT_1:def 7
    .=1/(r-s)*((1-s)*v2)-1/(r-s)*((1-r)*v1) by A4,RLVECT_1:34
    .=(1/(r-s)*(1-s))*v2-1/(r-s)*((1-r)*v1) by RLVECT_1:def 7
    .=(1/(r-s)*(1-s))*v2-(1/(r-s)*(1-r))*v1 by RLVECT_1:def 7
    .=(1/(r-s)*(1-s))*v2+-(1/(r-s)*(1-r))*v1 by RLVECT_1:def 11
    .=(1-(r-1)/(r-s))*v2+((r-1)/(r-s))*v1 by A7,A8,RLVECT_4:3;
   hence thesis by A1,A5,RUSUB_4:def 4;
  end;
  suppose A9: r=s;
   A10: 1-r<>0 by A2;
   (1-r)*v1=(1-r)*v2 by A3,A9,RLVECT_1:8;
   hence thesis by A9,A10,RLVECT_1:36;
  end;
 end;
