reserve p1,p2,p3,p4,p5,p6,p,pc for Point of TOP-REAL 2;
reserve a,b,c,r,s for Real;

theorem Th12:
  p in LSeg(p1,p3) & p in LSeg(p1,p4) & p3<>p4 & p<>p1 implies p3
  in LSeg(p1,p4) or p4 in LSeg(p1,p3)
proof
  assume p in LSeg(p1,p3);
  then consider l1 be Real such that
A1: p=(1-l1)*p1+l1*p3 and
A2: 0<=l1 and
  l1<=1;
  assume p in LSeg(p1,p4);
  then consider l2 be Real such that
A3: p=(1-l2)*p1+l2*p4 and
A4: 0<=l2 and
  l2<=1;
  (1+(-l1))*p1+l1*p3 = (1+(-l2))*p1+l2*p4 by A1,A3;
  then 1*p1+(-l1)*p1+l1*p3 = (1+(-l2))*p1+l2*p4 by RLVECT_1:def 6;
  then 1*p1+(-l1)*p1+l1*p3 = 1*p1+(-l2)*p1+l2*p4 by RLVECT_1:def 6;
  then p1+(-l1)*p1+l1*p3 = 1*p1+(-l2)*p1+l2*p4 by RLVECT_1:def 8;
  then p1+(-l1)*p1+l1*p3 = p1+(-l2)*p1+l2*p4 by RLVECT_1:def 8;
  then -p1+(p1+((-l1)*p1+l1*p3)) = -p1+(p1+(-l2)*p1+l2*p4) by RLVECT_1:def 3;
  then (-p1+p1)+((-l1)*p1+l1*p3) = -p1+(p1+(-l2)*p1+l2*p4) by RLVECT_1:def 3;
  then 0.TOP-REAL 2+((-l1)*p1+l1*p3) = -p1+(p1+(-l2)*p1+l2*p4) by RLVECT_1:5;
  then (-l1)*p1+l1*p3 = -p1+(p1+(-l2)*p1+l2*p4) by RLVECT_1:4;
  then (-l1)*p1+l1*p3 = -p1+(p1+((-l2)*p1+l2*p4)) by RLVECT_1:def 3;
  then (-l1)*p1+l1*p3 = (-p1+p1)+((-l2)*p1+l2*p4) by RLVECT_1:def 3;
  then (-l1)*p1+l1*p3 = 0.TOP-REAL 2+((-l2)*p1+l2*p4) by RLVECT_1:5;
  then
A5: (-l1)*p1+l1*p3 = (-l2)*p1+l2*p4 by RLVECT_1:4;
  assume that
A6: p3<>p4 and
A7: p<>p1;
  per cases;
  suppose
A8: l1<=l2;
    per cases by A8,XXREAL_0:1;
    suppose
A9:   l1<l2;
      (1/l2)*((-l1)*p1)+(1/l2)*(l1*p3)=(1/l2)*((-l2)*p1+l2*p4) by A5,
RLVECT_1:def 5;
      then (1/l2)*(-l1)*p1+(1/l2)*(l1*p3) = (1/l2)*((-l2)*p1+l2*p4) by
RLVECT_1:def 7;
      then (-1)*((1/l2)*l1)*p1+(1/l2)*l1*p3 = (1/l2)*((-l2)*p1+l2*p4) by
RLVECT_1:def 7;
      then (-1)*(l1/l2*1)*p1+(1/l2)*l1*p3 = (1/l2)*((-l2)*p1+l2*p4) by
XCMPLX_1:75;
      then (-l1/l2)*p1+(l1/l2)*1*p3 = (1/l2)*((-l2)*p1+l2*p4) by XCMPLX_1:75;
      then (-l1/l2)*p1+(l1/l2)*p3 = (1/l2)*((-l2)*p1)+(1/l2)*(l2*p4) by
RLVECT_1:def 5;
      then (-l1/l2)*p1+(l1/l2)*p3 = (1/l2)*(-l2)*p1+(1/l2)*(l2*p4) by
RLVECT_1:def 7;
      then (-l1/l2)*p1+(l1/l2)*p3 = (-1)*((1/l2)*l2)*p1+(1/l2)*l2*p4 by
RLVECT_1:def 7;
      then (-l1/l2)*p1+(l1/l2)*p3 = (-1)*((l2/l2)*1)*p1+(1/l2)*l2*p4 by
XCMPLX_1:75;
      then (-l1/l2)*p1+(l1/l2)*p3 = (-1)*(l2/l2)*p1+(l2/l2)*1*p4 by XCMPLX_1:75
;
      then (-l1/l2)*p1+(l1/l2)*p3 = (-1)*1*p1+(l2/l2)*p4 by A2,A9,XCMPLX_1:60;
      then (-l1/l2)*p1+(l1/l2)*p3 = (-1)*p1+1*p4 by A2,A9,XCMPLX_1:60;
      then (-l1/l2)*p1+(l1/l2)*p3 = -p1+1*p4;
      then
A10:  (-l1/l2)*p1+(l1/l2)*p3 = p4+(-p1) by RLVECT_1:def 8;
      l1/l2<l2/l2 by A2,A9,XREAL_1:74;
      then
A11:  l1/l2<1 by A2,A9,XCMPLX_1:60;
      p4 = p4 - p1 + p1 by RLVECT_4:1
        .= p1+(-l1/l2)*p1+(l1/l2)*p3 by A10,RLVECT_1:def 3
        .= 1*p1+(-l1/l2)*p1+(l1/l2)*p3 by RLVECT_1:def 8
        .= (1+(-l1/l2))*p1+(l1/l2)*p3 by RLVECT_1:def 6
        .= (1-l1/l2)*p1+(l1/l2)*p3;
      hence thesis by A2,A4,A11;
    end;
    suppose
      l1=l2;
      then l1*p1+(-l1)*p1+l1*p3 = l1*p1+((-l1)*p1+l1*p4) by A5,RLVECT_1:def 3;
      then (l1+(-l1))*p1+l1*p3 = l1*p1+((-l1)*p1+l1*p4) by RLVECT_1:def 6;
      then 0.TOP-REAL 2+l1*p3 = l1*p1+((-l1)*p1+l1*p4) by RLVECT_1:10;
      then l1*p3 = l1*p1+((-l1)*p1+l1*p4) by RLVECT_1:4;
      then l1*p3 = l1*p1+(-l1)*p1+l1*p4 by RLVECT_1:def 3;
      then l1*p3 = (l1+(-l1))*p1+l1*p4 by RLVECT_1:def 6;
      then l1*p3 = 0.TOP-REAL 2+l1*p4 by RLVECT_1:10;
      then
A12:  l1*p3 = l1*p4 by RLVECT_1:4;
      per cases by A12,RLVECT_1:36;
      suppose
        l1=0;
        then p = p1+(0 qua Real)*p3 by A1,RLVECT_1:def 8
          .= p1+0.TOP-REAL 2 by RLVECT_1:10
          .= p1 by RLVECT_1:4;
        hence thesis by A7;
      end;
      suppose
        p3=p4;
        hence thesis by A6;
      end;
    end;
  end;
  suppose
A13: l1>l2;
    (1/l1)*((-l1)*p1)+(1/l1)*(l1*p3) = (1/l1)*((-l2)*p1+l2*p4) by A5,
RLVECT_1:def 5;
    then (1/l1)*(-l1)*p1+(1/l1)*(l1*p3) = (1/l1)*((-l2)*p1+l2*p4) by
RLVECT_1:def 7;
    then (-1)*((1/l1)*l1)*p1+(1/l1)*l1*p3 = (1/l1)*((-l2)*p1+l2*p4) by
RLVECT_1:def 7;
    then (-1)*(l1/l1*1)*p1+(1/l1)*l1*p3 = (1/l1)*((-l2)*p1+l2*p4) by
XCMPLX_1:75;
    then (-l1/l1)*p1+(l1/l1)*1*p3 = (1/l1)*((-l2)*p1+l2*p4) by XCMPLX_1:75;
    then (-1)*p1+(l1/l1)*1*p3 = (1/l1)*((-l2)*p1+l2*p4) by A4,A13,XCMPLX_1:60;
    then (1)*p3+(-1)*p1 = (1/l1)*((-l2)*p1+l2*p4) by A4,A13,XCMPLX_1:60;
    then (1)*p3+-p1 = (1/l1)*((-l2)*p1+l2*p4);
    then p3+ -p1 = (1/l1)*((-l2)*p1+l2*p4) by RLVECT_1:def 8;
    then p3-p1 = (1/l1)*((-l2)*p1)+(1/l1)*(l2*p4) by RLVECT_1:def 5;
    then p3-p1 = (1/l1)*(-l2)*p1+(1/l1)*(l2*p4) by RLVECT_1:def 7;
    then p3-p1 = (-1)*((1/l1)*l2)*p1+(1/l1)*l2*p4 by RLVECT_1:def 7;
    then
A14: p3-p1 = (-1)*((l2/l1)*1)*p1+(1/l1)*l2*p4 by XCMPLX_1:75;
    l2/l1<l1/l1 by A4,A13,XREAL_1:74;
    then
A15: l2/l1<1 by A4,A13,XCMPLX_1:60;
    p3 = p3 - p1 + p1 by RLVECT_4:1
      .= (-l2/l1)*p1+(l2/l1)*p4 + p1 by A14,XCMPLX_1:75
      .= p1+(-l2/l1)*p1+(l2/l1)*p4 by RLVECT_1:def 3
      .= 1*p1+(-l2/l1)*p1+(l2/l1)*p4 by RLVECT_1:def 8
      .= (1+(-l2/l1))*p1+(l2/l1)*p4 by RLVECT_1:def 6
      .= (1-l2/l1)*p1+(l2/l1)*p4;
    hence thesis by A2,A4,A15;
  end;
end;
