reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;
reserve u,v for non zero Element of TOP-REAL 3;
reserve P,Q,R for POINT of IncProjSp_of real_projective_plane,
            L for LINE of IncProjSp_of real_projective_plane,
        p,q,r for Point of real_projective_plane;
reserve u,v,w for non zero Element of TOP-REAL 3;

theorem
  for P,Q,R being Element of absolute st P,Q,R are_mutually_distinct holds
  not P,Q,R are_collinear
  proof
    let P,Q,R be Element of absolute;
    assume
A1: P,Q,R are_mutually_distinct;
    consider up be Element of TOP-REAL 3 such that
A2: up is non zero and
A3: P = Dir up by ANPROJ_1:26;
    consider uq be Element of TOP-REAL 3 such that
A4: uq is non zero and
A5: Q = Dir uq by ANPROJ_1:26;
    consider ur be Element of TOP-REAL 3 such that
A6: ur is non zero and
A7: R = Dir ur by ANPROJ_1:26;
    reconsider up1 = up`1,up2 = up`2,up3 = up`3,
               uq1 = uq`1,uq2 = uq`2,uq3 = uq`3,
               ur1 = ur`1,ur2 = ur`2,ur3 = ur`3 as Real;
    up.3 <> 0 & uq.3 <> 0 & ur.3 <> 0 by A2,A3,A4,A5,A6,A7,Th67;
    then
A8: up3 <> 0 & uq3 <> 0 & ur3 <> 0 by EUCLID_5:def 3;
    reconsider vp1 = up1 / up3,vp2 = up2 / up3, vq1 = uq1 / uq3,
               vq2 = uq2 / uq3, vr1 = ur1 / ur3,vr2 = ur2 / ur3 as Real;
    reconsider vp = |[ vp1,vp2,1 ]|, vq = |[ vq1,vq2,1 ]|,
               vr = |[ vr1,vr2,1 ]| as non zero Element of TOP-REAL 3
                 by Th36;
A9: vp`3 = 1 & vq`3 = 1 & vr`3 = 1 by EUCLID_5:2;
A10: vp.3 = 1  & vq.3 = 1 & vr.3 = 1;
A11: P = Dir vp & Q = Dir vq & R = Dir vr
    proof
      thus P = Dir vp
      proof
        are_Prop up,vp
        proof
          up3 * vp = |[up3 * (up1/ up3),up3 * (up2 / up3),up3 * 1]|
                        by EUCLID_5:8
                  .= |[up1,up3 * (up2 / up3),up3 * 1]| by A8,XCMPLX_1:87
                  .= |[up1,up2,up3 * 1]| by A8,XCMPLX_1:87
                  .= up by EUCLID_5:3;
          hence thesis by A8,ANPROJ_1:1;
        end;
        hence thesis by A2,A3,ANPROJ_1:22;
      end;
      thus Q = Dir vq
      proof
        are_Prop uq,vq
        proof
          uq3 * vq = |[uq3 * (uq1/ uq3),uq3 * (uq2 / uq3),uq3 * 1]|
            by EUCLID_5:8
                  .= |[uq1,uq3 * (uq2 / uq3),uq3 * 1]| by A8,XCMPLX_1:87
                  .= |[uq1,uq2,uq3 * 1]| by A8,XCMPLX_1:87
                  .= uq by EUCLID_5:3;
          hence thesis by A8,ANPROJ_1:1;
        end;
        hence thesis by A4,A5,ANPROJ_1:22;
      end;
      thus R = Dir vr
      proof
        are_Prop ur,vr
        proof
          ur3 * vr = |[ur3 * (ur1/ ur3),ur3 * (ur2 / ur3),ur3 * 1]|
            by EUCLID_5:8
          .= |[ur1,ur3 * (ur2 / ur3),ur3 * 1]| by A8,XCMPLX_1:87
          .= |[ur1,ur2,ur3 * 1]| by A8,XCMPLX_1:87
          .= ur by EUCLID_5:3;
          hence thesis by A8,ANPROJ_1:1;
        end;
        hence thesis by A6,A7,ANPROJ_1:22;
      end;
    end;
    assume P,Q,R are_collinear;
    then consider tp,tq,tr be Element of TOP-REAL 3 such that
A12: P = Dir(tp) & Q = Dir(tq) & R = Dir(tr) and
A13: tp is not zero & tq is not zero & tr is not zero and
A14: ex a1,b1,c1 be Real st a1*tp + b1*tq + c1*tr = 0.TOP-REAL 3 &
      (a1<>0 or b1<>0 or c1<>0) by ANPROJ_8:11;
    consider a1,b1,c1 be Real such that
A15: a1 * tp + b1 * tq + c1 * tr = 0.TOP-REAL 3 and
A16: (a1<>0 or b1<>0 or c1<>0) by A14;
A17: are_Prop tp,vp & are_Prop tq,vq & are_Prop tr,vr
      by A12,A13,A11,ANPROJ_1:22;
    consider lp be Real such that
A18: lp <> 0 and
A19: tp = lp * vp by A17,ANPROJ_1:1;
    consider lq be Real such that
A20: lq <> 0 and
A21: tq = lq * vq by A17,ANPROJ_1:1;
    consider lr be Real such that
A22: lr <> 0 and
A23: tr = lr * vr by A17,ANPROJ_1:1;
    reconsider A = |[vp`1,vp`2]|, B = |[vq`1,vq`2]|,
               C = |[vr`1,vr`2]| as Element of TOP-REAL 2;
A24: B.1 = B`1 by EUCLID:def 9
          .= vq`1 by EUCLID:52
          .= vq.1 by EUCLID_5:def 1;
A25: B.2 = B`2 by EUCLID:def 10
          .= vq`2 by EUCLID:52
          .= vq.2 by EUCLID_5:def 2;
A26: C.1 = C`1 by EUCLID:def 9
          .= vr`1 by EUCLID:52
          .= vr.1 by EUCLID_5:def 1;
A27: C.2 = C`2 by EUCLID:def 10
          .= vr`2 by EUCLID:52
          .= vr.2 by EUCLID_5:def 2;
A28: A.1 = A`1 by EUCLID:def 9
          .= vp`1 by EUCLID:52
          .= vp.1 by EUCLID_5:def 1;
A29: A.2 = A`2 by EUCLID:def 10
          .= vp`2 by EUCLID:52
          .= vp.2 by EUCLID_5:def 2;
A30: A <> B
     proof
       assume A = B;
       then
A30BIS:  vp`1 = vq`1 & vp`2 = vq`2 by FINSEQ_1:77;
       vp = |[vp`1,vp`2,vp`3]| by EUCLID_5:3
              .= vq by A9,A30BIS,EUCLID_5:3;
       hence contradiction by A11,A1;
     end;
A31: A <> C
     proof
       assume A = C;
       then
A31BIS:  vp`1 = vr`1 & vp`2 = vr`2 by FINSEQ_1:77;
       vp = |[vp`1,vp`2,vp`3]| by EUCLID_5:3
              .= vr by A9,A31BIS,EUCLID_5:3;
       hence contradiction by A11,A1;
     end;
A32: B <> C
     proof
       assume B = C;
       then
A32BIS:  vq`1 = vr`1 & vq`2 = vr`2 by FINSEQ_1:77;
       vq = |[vq`1,vq`2,vq`3]| by EUCLID_5:3
              .= vr by A9,A32BIS,EUCLID_5:3;
       hence contradiction by A11,A1;
     end;
A34: A in Sphere(0.TOP-REAL 2,1)
     proof
A35:   qfconic(1,1,-1,0,0,0,vp) = 0
       proof
         P in conic(1,1,-1,0,0,0);
         then P in {P where P is Point of ProjectiveSpace TOP-REAL 3:
           for u being Element of TOP-REAL 3 st u is non zero & P = Dir u holds
           qfconic(1,1,-1,0,0,0,u) = 0} by PASCAL:def 2;
         then ex PP be Point of ProjectiveSpace TOP-REAL 3 st P = PP & for u
           being Element of TOP-REAL 3 st u is non zero & PP = Dir u holds
           qfconic(1,1,-1,0,0,0,u) = 0;
         hence thesis by A11;
       end;
       vp.1 = vp`1 & vp.2 = vp`2 by EUCLID_5:def 1,def 2;
       hence A in Sphere(0.TOP-REAL 2,1) by A35,A10,Th70;
     end;
A36: qfconic(1,1,-1,0,0,0,vq) = 0
     proof
       Q in conic(1,1,-1,0,0,0);
       then Q in {P where P is Point of ProjectiveSpace TOP-REAL 3:
         for u being Element of TOP-REAL 3 st u is non zero & P = Dir u holds
         qfconic(1,1,-1,0,0,0,u) = 0} by PASCAL:def 2;
       then ex QP be Point of ProjectiveSpace TOP-REAL 3 st
         Q = QP & for u being Element of TOP-REAL 3 st u is non zero &
         QP = Dir u holds qfconic(1,1,-1,0,0,0,u) = 0;
       hence thesis by A11;
     end;
     vq.1 = vq`1 & vq.2 = vq`2 by EUCLID_5:def 1,def 2; then
A37: B in Sphere(0.TOP-REAL 2,1) by A36,A10,Th70;
A38: C in Sphere(0.TOP-REAL 2,1)
     proof
A39:   qfconic(1,1,-1,0,0,0,vr) = 0
       proof
         R in conic(1,1,-1,0,0,0);
         then R in {P where P is Point of ProjectiveSpace TOP-REAL 3:
           for u being Element of TOP-REAL 3 st u is non zero & P = Dir u holds
           qfconic(1,1,-1,0,0,0,u) = 0} by PASCAL:def 2;
         then ex RP be Point of ProjectiveSpace TOP-REAL 3 st R = RP &
           for u being Element of TOP-REAL 3 st u is non zero &
             RP = Dir u holds
            qfconic(1,1,-1,0,0,0,u) = 0;
         hence thesis by A11;
       end;
       vr.1 = vr`1 & vr.2 = vr`2 by EUCLID_5:def 1,def 2;
       hence C in Sphere(0.TOP-REAL 2,1) by A39,A10,Th70;
    end;
A40: halfline(B,C) /\ Sphere(0.TOP-REAL 2,1) = {B,C} &
      halfline(C,B) /\ Sphere(0.TOP-REAL 2,1) = {C,B} &
      halfline(A,C) /\ Sphere(0.TOP-REAL 2,1) = {A,C} &
      halfline(C,A) /\ Sphere(0.TOP-REAL 2,1) = {C,A} &
      halfline(A,B) /\ Sphere(0.TOP-REAL 2,1) = {A,B} &
      halfline(B,A) /\ Sphere(0.TOP-REAL 2,1) = {B,A}
        by A37,A38,A34,TOPREAL9:36;
      per cases by A16;
      suppose a1 <> 0; then
A41:    lp * vp = ((-b1)/a1) * (lq * vq) + ((-c1)/a1) * (lr * vr)
          by A19,A21,A23,A15,ANPROJ_8:12
               .= ((-b1)/a1)*lq * vq + ((-c1)/a1) * (lr * vr) by RVSUM_1:49
               .= ((-b1)/a1)*lq * vq + ((-c1)/a1)*lr * vr by RVSUM_1:49;
      reconsider m1 = 1/lp * (((-b1)/a1) * lq),
                 m2 = 1/lp * (((-c1)/a1)* lr) as Real;
      1 = lp / lp by A18,XCMPLX_1:60
       .= 1/lp * lp by XCMPLX_1:99; then
A42:  vp = (1/lp * lp) * vp by RVSUM_1:52
         .= 1/lp * (((-b1)/a1)*lq * vq + ((-c1)/a1)*lr * vr) by A41,RVSUM_1:49
         .= 1/lp * (((-b1)/a1)*lq * vq) + 1/lp * (((-c1)/a1)*lr * vr)
            by RVSUM_1:51
         .= 1/lp * (((-b1)/a1)*lq) * vq + 1/lp * (((-c1)/a1)*lr * vr)
            by RVSUM_1:49
         .= m1 * vq + m2 * vr by RVSUM_1:49;
A43:  m1 = 1 - m2
      proof
        m1 * vq + m2 * vr = |[m1 * vq`1,m1 * vq`2,m1 * vq`3]| + m2 * vr
                             by EUCLID_5:7
                         .= |[m1 * vq`1,m1 * vq`2,m1 * vq`3]|
                             + |[m2 * vr`1,m2 * vr`2,m2 * vr`3]| by EUCLID_5:7
                         .= |[m1 * vq`1 + m2 * vr`1,
                              m1 * vq`2 + m2 * vr`2,
                              m1 * vq`3 + m2 * vr`3]| by EUCLID_5:6;
        then vp`3 = m1 * vq`3 + m2 * vr`3 by A42,EUCLID_5:14;
        hence thesis by A9;
      end;
      per cases;
      suppose
A44:    0 <= m2;
A45:    A in halfline(B,C)
        proof
          reconsider tu = (1-m2)*vq, tv = m2*vr as Element of TOP-REAL 3;
A46:      ((1 - m2) * vq).1 = (1-m2) * vq.1 by RVSUM_1:44;
A47:      tu`1 = tu.1 & tv`1 = tv.1 by EUCLID_5:def 1;
A48:      A`1 = vp`1 by EUCLID:52
             .= |[tu`1+tv`1,tu`2+tv`2,tu`3+tv`3]|`1 by A43,A42,EUCLID_5:5
             .= tu.1 + tv.1 by A47,EUCLID_5:2
             .= (1-m2) * B.1 + m2 * C.1 by A46,RVSUM_1:44,A24,A26;
A49:      ((1 - m2) * vq).2 = (1-m2) * vq.2 by RVSUM_1:44;
A50:      tu`2 = tu.2 & tv`2 = tv.2 by EUCLID_5:def 2;
A51:      A`2 = vp`2 by EUCLID:52
             .= |[tu`1+tv`1,tu`2+tv`2,tu`3+tv`3]|`2 by A43,A42,EUCLID_5:5
             .= tu.2 + tv.2 by A50,EUCLID_5:2
             .= (1-m2) * B.2 + m2 * C.2 by A49,A25,A27,RVSUM_1:44;
          A = |[(1-m2)*B.1 + m2 * C.1,(1-m2)*B.2+m2 * C.2]|
              by A48,A51,EUCLID:53
           .= |[(1-m2) * B.1,(1-m2)*B.2]| + |[m2 * C.1,m2 * C.2]| by EUCLID:56
           .= (1 - m2) * |[B.1,B.2]| + |[m2 * C.1,m2 * C.2]| by EUCLID:58
           .= (1 - m2) * B + m2 * C by EUCLID:58;
          hence thesis by A44,TOPREAL9:26;
        end;
        A in halfline(B,C) /\ Sphere(0.TOP-REAL 2,1)
          by A45,A34,XBOOLE_0:def 4;
        hence contradiction by A30,A31,A40,TARSKI:def 2;
      end;
      suppose
A54:    m2 < 0;
        set m3 = 1 - m2;
        A in halfline(C,B)
        proof
          reconsider tu = (1-m3)*vr, tv = m3*vq as Element of TOP-REAL 3;
A55:      ((1 - m3) * vr).1 = (1-m3) * vr.1 by RVSUM_1:44;
A56:      tu`1 = tu.1 & tv`1 = tv.1 by EUCLID_5:def 1;
A56b:     A`1 = vp`1 by EUCLID:52
             .= |[tu`1+tv`1,tu`2+tv`2,tu`3+tv`3]|`1 by A43,A42,EUCLID_5:5
             .= tu.1 + tv.1 by EUCLID_5:2,A56
             .= (1-m3) * C.1 + m3 * B.1 by A24,A26,A55,RVSUM_1:44;
A57:      ((1 - m3) * vr).2 = (1-m3) * vr.2 by RVSUM_1:44;
A58:      tu`2 = tu.2 & tv`2 = tv.2 by EUCLID_5:def 2;
A59:      A`2 = vp`2 by EUCLID:52
             .= |[tu`1+tv`1,tu`2+tv`2,tu`3+tv`3]|`2 by A43,A42,EUCLID_5:5
             .= tu`2 + tv`2 by EUCLID_5:2
             .= (1-m3) * C.2 + m3 * B.2 by A57,A58,RVSUM_1:44,A25,A27;
          A = |[(1-m3)*C.1 + m3 * B.1,(1-m3)*C.2+m3 * B.2]|
            by A56b,A59,EUCLID:53
           .= |[(1-m3) * C.1,(1-m3)*C.2]| + |[m3 * B.1,m3 * B.2]| by EUCLID:56
           .= (1 - m3) * |[C.1,C.2]| + |[m3 * B.1,m3 * B.2]| by EUCLID:58
           .= (1 - m3) * C + m3 * B by EUCLID:58;
          hence thesis by A54,TOPREAL9:26;
        end;
        then A in {C,B} by A40,A34,XBOOLE_0:def 4;
        hence contradiction by A30,A31,TARSKI:def 2;
       end;
    end;
    suppose
A62:  b1 <> 0;
      b1 * tq + a1 * tp + c1 * tr = 0.TOP-REAL 3 by A15;
      then
A63:  lq * vq = ((-a1)/b1) * (lp * vp) + ((-c1)/b1) * (lr * vr)
      by A19,A21,A23,A62,ANPROJ_8:12
             .= ((-a1)/b1)*lp * vp + ((-c1)/b1) * (lr * vr) by RVSUM_1:49
             .= ((-a1)/b1)*lp * vp + ((-c1)/b1)*lr * vr by RVSUM_1:49;
      reconsider m1 = 1/lq * (((-a1)/b1) * lp), m2 = 1/lq * (((-c1)/b1)* lr)
        as Real;
      1 = lq / lq by A20,XCMPLX_1:60
       .= 1/lq * lq by XCMPLX_1:99; then
A65:  vq = (1/lq * lq) * vq by RVSUM_1:52
         .= 1/lq * (((-a1)/b1)*lp * vp + ((-c1)/b1)*lr * vr) by A63,RVSUM_1:49
         .= 1/lq * (((-a1)/b1)*lp * vp) + 1/lq * (((-c1)/b1)*lr * vr)
            by RVSUM_1:51
         .= 1/lq * (((-a1)/b1)*lp) * vp + 1/lq * (((-c1)/b1)*lr * vr)
            by RVSUM_1:49
         .= m1 * vp + m2 * vr by RVSUM_1:49;
A66:  m1 = 1 - m2
      proof
        m1 * vp + m2 * vr = |[m1 * vp`1,m1 * vp`2,
                              m1 * vp`3]| + m2 * vr by EUCLID_5:7
                         .= |[m1 * vp`1,m1 * vp`2,m1 * vp`3]|
                             + |[m2 * vr`1,m2 * vr`2,m2 * vr`3]| by EUCLID_5:7
                         .= |[m1 * vp`1 + m2 * vr`1, m1 * vp`2 + m2 * vr`2,
                              m1 * vp`3 + m2 * vr`3]| by EUCLID_5:6;
        then 1 = m1 * 1 + m2 * 1 by A9,A65,EUCLID_5:14;
        hence thesis;
      end;
      per cases;
      suppose
A67:    0 <= m2;
        B in halfline(A,C)
        proof
          reconsider tu = (1-m2)*vp, tv = m2*vr as Element of TOP-REAL 3;
A68:      ((1 - m2) * vp).1 = (1-m2) * vp.1 by RVSUM_1:44;
A69:      tu`1 = tu.1 & tv`1 = tv.1 by EUCLID_5:def 1;
A70:      B`1 = vq`1 by EUCLID:52
             .= |[tu`1+tv`1,tu`2+tv`2,tu`3+tv`3]|`1 by A66,A65,EUCLID_5:5
             .= tu`1 + tv`1 by EUCLID_5:2
             .= (1-m2) * A.1 + m2 * C.1 by A26,A28,A68,A69,RVSUM_1:44;
A71:      ((1 - m2) * vp).2 = (1-m2) * vp.2 by RVSUM_1:44;
A72:      tu`2 = tu.2 & tv`2 = tv.2 by EUCLID_5:def 2;
A73:      B`2 = vq`2 by EUCLID:52
             .= |[tu`1+tv`1,tu`2+tv`2,tu`3+tv`3]|`2 by A66,A65,EUCLID_5:5
             .= tu`2 + tv`2 by EUCLID_5:2
             .= (1-m2) * A.2 + m2 * C.2 by A27,A29,A71,A72,RVSUM_1:44;
          B = |[(1-m2)*A.1 + m2 * C.1,(1-m2)*A.2+m2 * C.2]|
              by A70,A73,EUCLID:53
           .= |[(1-m2) * A.1,(1-m2)*A.2]| + |[m2 * C.1,m2 * C.2]| by EUCLID:56
           .= (1 - m2) * |[A.1,A.2]| + |[m2 * C.1,m2 * C.2]| by EUCLID:58
           .= (1 - m2) * A + m2 * C by EUCLID:58;
          hence thesis by A67,TOPREAL9:26;
        end;
        then B in {A,C} by A40,A37,XBOOLE_0:def 4;
        hence contradiction by A30,A32,TARSKI:def 2;
      end;
      suppose
A76:    m2 < 0;
        set m3 = 1 - m2;
        B in halfline(C,A)
        proof
          reconsider tu = (1-m3)*vr, tv = m3*vp as Element of TOP-REAL 3;
A77:      ((1 - m3) * vr).1 = (1-m3) * vr.1 by RVSUM_1:44;
A78:      tu`1 = tu.1 & tv`1 = tv.1 by EUCLID_5:def 1;
A79:      B`1 = vq`1 by EUCLID:52
             .= |[tu`1+tv`1,tu`2+tv`2,tu`3+tv`3]|`1 by EUCLID_5:5,A66,A65
             .= tu`1 + tv`1 by EUCLID_5:2
             .= (1-m3) * C.1 + m3 * A.1 by A26,A28,A77,A78,RVSUM_1:44;
A80:      ((1 - m3) * vr).2 = (1-m3) * vr.2 by RVSUM_1:44;
A81:      tu`2 = tu.2 & tv`2 = tv.2 by EUCLID_5:def 2;
A82:      B`2 = vq`2 by EUCLID:52
             .= |[tu`1+tv`1,tu`2+tv`2,tu`3+tv`3]|`2 by A66,A65,EUCLID_5:5
             .= tu`2 + tv`2 by EUCLID_5:2
             .= (1-m3) * C.2 + m3 * A.2 by A27,A29,A80,A81,RVSUM_1:44;
          B = |[(1-m3)*C.1 + m3 * A.1,(1-m3)*C.2+m3 * A.2]|
             by A79,A82,EUCLID:53
           .= |[(1-m3) * C.1,(1-m3)*C.2]| + |[m3 * A.1,m3 * A.2]| by EUCLID:56
           .= (1 - m3) * |[C.1,C.2]| + |[m3 * A.1,m3 * A.2]| by EUCLID:58
           .= (1 - m3) * C + m3 * A by EUCLID:58;
          hence thesis by A76,TOPREAL9:26;
        end;
        then B in {C,A} by A40,A37,XBOOLE_0:def 4;
        hence contradiction by A30,A32,TARSKI:def 2;
       end;
     end;
      suppose
A85:    c1 <> 0;
        c1 * tr + a1 * tp + b1 * tq = 0.TOP-REAL 3 by A15,RVSUM_1:15;
        then
A86:    lr * vr = ((-a1)/c1) * (lp * vp) + ((-b1)/c1) * (lq * vq)
          by A19,A21,A23,A85,ANPROJ_8:12
               .= ((-a1)/c1)*lp * vp + ((-b1)/c1) * (lq * vq) by RVSUM_1:49
               .= ((-a1)/c1)*lp * vp + ((-b1)/c1)* lq * vq by RVSUM_1:49;
        reconsider m1 = 1/lr * (((-a1)/c1) * lp),
                   m2 = 1/lr * (((-b1)/c1)* lq) as Real;
        1 = lr / lr by A22,XCMPLX_1:60
         .= 1/lr * lr by XCMPLX_1:99; then
A87:    vr = (1/lr * lr) * vr by RVSUM_1:52
          .= 1/lr * (((-a1)/c1)*lp * vp + ((-b1)/c1)*lq * vq) by A86,RVSUM_1:49
          .= 1/lr * (((-a1)/c1)*lp * vp) + 1/lr * (((-b1)/c1)*lq * vq)
             by RVSUM_1:51
          .= 1/lr * (((-a1)/c1)*lp) * vp + 1/lr * (((-b1)/c1)*lq * vq)
             by RVSUM_1:49
          .= m1 * vp + m2 * vq by RVSUM_1:49;
A88:    m1 = 1 - m2
        proof
          m1 * vp + m2 * vq = |[m1 * vp`1,m1 * vp`2,m1 * vp`3]| + m2 * vq
                                by EUCLID_5:7
                           .= |[m1 * vp`1,m1 * vp`2,m1 * vp`3]|
                              + |[m2 * vq`1,m2 * vq`2,m2 * vq`3]| by EUCLID_5:7
                           .= |[m1 * vp`1 + m2 * vq`1,
                                m1 * vp`2 + m2 * vq`2,
                                m1 * vp`3 + m2 * vq`3]| by EUCLID_5:6;
        then 1 = m1 * 1 + m2 * 1 by A87,EUCLID_5:14,A9;
        hence thesis;
      end;
      per cases;
      suppose
A89:    0 <= m2;
        C in halfline(A,B)
        proof
          reconsider tu = (1-m2)*vp, tv = m2*vq as Element of TOP-REAL 3;
A90:      ((1 - m2) * vp).1 = (1-m2) * vp.1 by RVSUM_1:44;
A91:      tu`1 = tu.1 & tv`1 = tv.1 by EUCLID_5:def 1;
A92:      C`1 = vr`1 by EUCLID:52
             .= |[tu`1+tv`1,tu`2+tv`2,tu`3+tv`3]|`1 by A88,A87,EUCLID_5:5
             .= tu.1 + tv.1 by A91,EUCLID_5:2
             .= (1-m2) * A.1 + m2 * B.1 by A24,A28,A90,RVSUM_1:44;
A93:      ((1 - m2) * vp).2 = (1-m2) * vp.2 by RVSUM_1:44;
A94:      tu`2 = tu.2 & tv`2 = tv.2 by EUCLID_5:def 2;
A95:      C`2 = vr`2 by EUCLID:52
             .= |[tu`1+tv`1,tu`2+tv`2,tu`3+tv`3]|`2 by A88,A87,EUCLID_5:5
             .= tu`2 + tv`2 by EUCLID_5:2
             .= (1-m2) * A.2 + m2 * B.2 by A25,A29,A93,A94,RVSUM_1:44;
          C = |[C`1,C`2]| by EUCLID:53
           .= |[(1-m2) * A.1,(1-m2)*A.2]| + |[m2 * B.1,m2 * B.2]|
                by A92,A95,EUCLID:56
            .= (1 - m2) * |[A.1,A.2]| + |[m2 * B.1,m2 * B.2]| by EUCLID:58
            .= (1 - m2) * A + m2 * B by EUCLID:58;
          hence thesis by A89,TOPREAL9:26;
        end;
        then C in {A,B} by A40,A38,XBOOLE_0:def 4;
        hence contradiction by A31,A32,TARSKI:def 2;
      end;
      suppose
A98:    m2 < 0;
        set m3 = 1 - m2;
        C in halfline(B,A)
        proof
          reconsider tu = (1-m3)*vq, tv = m3*vp as Element of TOP-REAL 3;
A99:      ((1 - m3) * vq).1 = (1-m3) * vq.1 by RVSUM_1:44;
A100:     tu`1 = tu.1 & tv`1 = tv.1 by EUCLID_5:def 1;
A101:     C`1 = vr`1 by EUCLID:52
             .= |[tu`1+tv`1,tu`2+tv`2,tu`3+tv`3]|`1 by A88,A87,EUCLID_5:5
             .= tu.1 + tv.1 by A100,EUCLID_5:2
             .= (1-m3) * B.1 + m3 * A.1 by A24,A28,A99,RVSUM_1:44;
             A102:     ((1 - m3) * vq).2 = (1-m3) * vq.2 by RVSUM_1:44;
A103:     tu`2 = tu.2 & tv`2 = tv.2 by EUCLID_5:def 2;
A104:     C`2 = vr`2 by EUCLID:52
             .= |[tu`1+tv`1,tu`2+tv`2,tu`3+tv`3]|`2 by A88,A87,EUCLID_5:5
             .= tu`2 + tv`2 by EUCLID_5:2
             .= (1-m3) * B.2 + m3 * A.2 by A25,A29,A102,A103,RVSUM_1:44;
          C = |[(1-m3)*B.1 + m3 * A.1,(1-m3)*B.2+m3 * A.2]|
            by A101,A104,EUCLID:53
           .= |[(1-m3) * B.1,(1-m3)*B.2]| + |[m3 * A.1,m3 * A.2]| by EUCLID:56
           .= (1 - m3) * |[B.1,B.2]| + |[m3 * A.1,m3 * A.2]| by EUCLID:58
           .= (1 - m3) * B + m3 * A by EUCLID:58;
          hence thesis by A98,TOPREAL9:26;
        end;
        then C in {B,A} by A40,A38,XBOOLE_0:def 4;
        hence contradiction by A31,A32,TARSKI:def 2;
       end;
    end;
  end;
