reserve r,lambda for Real,
  i,j,n for Nat;
reserve p,p1,p2,q1,q2 for Point of TOP-REAL 2,
  P, P1 for Subset of TOP-REAL 2;
reserve T for TopSpace;

theorem Th13:
  LSeg(|[ 0,0 ]|, |[ 0,1 ]|) = { p1 : p1`1 = 0 & p1`2 <= 1 & p1`2
>= 0} & LSeg(|[ 0,1 ]|, |[ 1,1 ]|) = { p2 : p2`1 <= 1 & p2`1 >= 0 & p2`2 = 1} &
LSeg(|[ 0,0 ]|, |[ 1,0 ]|) = { q1 : q1`1 <= 1 & q1`1 >= 0 & q1`2 = 0} & LSeg(|[
  1,0 ]|, |[ 1,1 ]|) = { q2 : q2`1 = 1 & q2`2 <= 1 & q2`2 >= 0}
proof
  set p0 = |[ 0,0 ]|, p01 = |[ 0,1 ]|, p10 = |[ 1,0 ]|, p1 = |[ 1,1 ]|;
  set L1 = { p : p`1 = 0 & p`2 <= 1 & p`2 >= 0}, L2 = { p : p`1 <= 1 & p`1 >=
0 & p`2 = 1}, L3 = { p : p`1 <= 1 & p`1 >= 0 & p`2 = 0}, L4 = { p : p`1 = 1 & p
  `2 <= 1 & p`2 >= 0};
A1: p01`2 = 1 by EUCLID:52;
A2: p01`1 = 0 by EUCLID:52;
A3: LSeg(p0,p01) c= L1
  proof
    let a be object;
    assume a in LSeg(p0,p01);
    then consider lambda such that
A4: a = (1-lambda)*p0 + lambda*p01 and
A5: 0 <= lambda and
A6: lambda <= 1;
    set q = (1-lambda)*p0 + lambda*p01;
A7: (1-lambda)*p0 + lambda*p01 = 0.TOP-REAL 2 + lambda*p01 by EUCLID:54
,RLVECT_1:10
      .= lambda*p01 by RLVECT_1:4
      .= |[lambda*(p01`1), lambda*(p01`2) ]| by EUCLID:57
      .= |[ 0, lambda ]| by A2,A1;
    then
A8: q`2 >= 0 by A5,EUCLID:52;
A9: q`1 = 0 by A7,EUCLID:52;
    q`2 <= 1 by A6,A7,EUCLID:52;
    hence thesis by A4,A9,A8;
  end;
  L1 c= LSeg(p0,p01)
  proof
    let a be object;
    assume a in L1;
    then consider p such that
A10: a = p and
A11: p`1 = 0 and
A12: p`2 <= 1 and
A13: p`2 >= 0;
    set lambda = p`2;
    (1-lambda)*p0 + lambda*p01 = 0.TOP-REAL 2 + lambda*p01 by EUCLID:54
,RLVECT_1:10
      .= lambda*p01 by RLVECT_1:4
      .= |[lambda*p01`1, lambda*p01`2 ]| by EUCLID:57
      .= p by A2,A1,A11,EUCLID:53;
    hence thesis by A10,A12,A13;
  end;
  hence L1 = LSeg(p0,p01) by A3;
A14: p1`2 = 1 by EUCLID:52;
A15: p1`1 = 1 by EUCLID:52;
A16: LSeg(p01,p1) c= L2
  proof
    let a be object;
    assume a in LSeg(p01,p1);
    then consider lambda such that
A17: a = (1-lambda)*p01 + lambda*p1 and
A18: 0 <= lambda and
A19: lambda <= 1;
    set q = (1-lambda)*p01 + lambda*p1;
A20: (1-lambda)*p01 + lambda*p1 = |[(1-lambda)*0, (1-lambda)*p01`2]| +
    lambda*p1 by A2,EUCLID:57
      .= |[0, 1-lambda]| + |[lambda, lambda*1]| by A1,A15,A14,EUCLID:57
      .= |[0+lambda, 1-lambda+lambda]| by EUCLID:56
      .= |[lambda, 1]|;
    then
A21: q`1 >= 0 by A18,EUCLID:52;
A22: q`2 = 1 by A20,EUCLID:52;
    q`1 <= 1 by A19,A20,EUCLID:52;
    hence thesis by A17,A21,A22;
  end;
  L2 c= LSeg(p01,p1)
  proof
    let a be object;
    assume a in L2;
    then consider p such that
A23: a = p and
A24: p`1 <= 1 and
A25: p`1 >= 0 and
A26: p`2 = 1;
    set lambda = p`1;
    (1-lambda)*p01 + lambda*p1 = |[(1-lambda)*0, (1-lambda)*p01`2]| +
    lambda*p1 by A2,EUCLID:57
      .= |[0, 1-lambda]| + |[lambda*1, lambda]| by A1,A15,A14,EUCLID:57
      .= |[0+lambda, 1-lambda+lambda]| by EUCLID:56
      .= p by A26,EUCLID:53;
    hence thesis by A23,A24,A25;
  end;
  hence L2 = LSeg(p01,p1) by A16;
A27: p10`2 = 0 by EUCLID:52;
A28: p10`1 = 1 by EUCLID:52;
A29: LSeg(p0,p10) c= L3
  proof
    let a be object;
    assume a in LSeg(p0,p10);
    then consider lambda such that
A30: a = (1-lambda)*p0 + lambda*p10 and
A31: 0 <= lambda and
A32: lambda <= 1;
    set q =(1-lambda)*p0 + lambda*p10;
A33: (1-lambda)*p0 + lambda*p10 = 0.TOP-REAL 2 + lambda*p10 by EUCLID:54
,RLVECT_1:10
      .= lambda*p10 by RLVECT_1:4
      .= |[ lambda*(p10`1), lambda*(p10`2) ]| by EUCLID:57
      .= |[ lambda, 0 ]| by A28,A27;
    then
A34: q`1 >= 0 by A31,EUCLID:52;
A35: q`2 = 0 by A33,EUCLID:52;
    q`1 <= 1 by A32,A33,EUCLID:52;
    hence thesis by A30,A34,A35;
  end;
A36: LSeg(p10,p1) c= L4
  proof
    let a be object;
    assume a in LSeg(p10,p1);
    then consider lambda such that
A37: a = (1-lambda)*p10 + lambda*p1 and
A38: 0 <= lambda and
A39: lambda <= 1;
    set q = (1-lambda)*p10 + lambda*p1;
A40: (1-lambda)*p10 + lambda*p1 = |[(1-lambda)*1, (1-lambda)*p10`2]| +
    lambda*p1 by A28,EUCLID:57
      .= |[(1-lambda), 0]| + |[lambda, lambda*1]| by A15,A14,A27,EUCLID:57
      .= |[1-lambda+lambda, 0+lambda]| by EUCLID:56
      .= |[1, lambda]|;
    then
A41: q`2 >= 0 by A38,EUCLID:52;
A42: q`1 = 1 by A40,EUCLID:52;
    q`2 <= 1 by A39,A40,EUCLID:52;
    hence thesis by A37,A42,A41;
  end;
  L3 c= LSeg(p0,p10)
  proof
    let a be object;
    assume a in L3;
    then consider p such that
A43: a = p and
A44: p`1 <= 1 and
A45: p`1 >= 0 and
A46: p`2 = 0;
    set lambda = p`1;
    (1-lambda)*p0 + lambda*p10 = 0.TOP-REAL 2 + lambda*p10 by EUCLID:54
,RLVECT_1:10
      .= lambda*p10 by RLVECT_1:4
      .= |[lambda*p10`1, lambda*p10`2 ]| by EUCLID:57
      .= p by A28,A27,A46,EUCLID:53;
    hence thesis by A43,A44,A45;
  end;
  hence L3 = LSeg(p0,p10) by A29;
  L4 c= LSeg(p10,p1)
  proof
    let a be object;
    assume a in L4;
    then consider p such that
A47: a = p and
A48: p`1 = 1 and
A49: p`2 <= 1 and
A50: p`2 >= 0;
    set lambda = p`2;
    (1-lambda)*p10 + lambda*p1 = |[(1-lambda)*1, (1-lambda)*p10`2]| +
    lambda*p1 by A28,EUCLID:57
      .= |[(1-lambda), 0]| + |[lambda*1, lambda]| by A15,A14,A27,EUCLID:57
      .= |[1-lambda+lambda, 0+lambda]| by EUCLID:56
      .= p by A48,EUCLID:53;
    hence thesis by A47,A49,A50;
  end;
  hence L4 = LSeg(p10,p1) by A36;
end;
