reserve p,p1,p2,p3,p11,p22,q,q1,q2 for Point of TOP-REAL 2,
  f,h for FinSequence of TOP-REAL 2,
  r,r1,r2,s,s1,s2 for Real,
  u,u1,u2,u5 for Point of Euclid 2,
  n,m,i,j,k for Nat,
  N for Nat,
  x,y,z for set;
reserve lambda for Real;

theorem
  f = <* p1,p2,p3 *> implies L~f = LSeg(p1,p2) \/ LSeg(p2,p3)
proof
  set M = {LSeg(f,i) : 1 <= i & i+1 <= len f };
  assume
A1: f = <* p1,p2,p3 *>;
  then
A2: len f = 2+1 by FINSEQ_1:45;
A3: f/.3=p3 by A1,FINSEQ_4:18;
A4: f/.1=p1 by A1,FINSEQ_4:18;
A5: f/.2=p2 by A1,FINSEQ_4:18;
A6: 1+1 in dom f by A1,Th1;
  M = {LSeg(p1,p2),LSeg(p2,p3)}
  proof
    thus M c= {LSeg(p1,p2),LSeg(p2,p3)}
    proof
      let x be object;
      assume x in M;
      then consider j such that
A7:   x = LSeg(f,j) and
A8:   1<=j and
A9:   j+1<=len f;
  j <= 2 by A2,A9,XREAL_1:6;
      then j = 0 or ... or j = 2;
      then per cases by A8;
      suppose
        j=1;
        then x = LSeg(p1,p2) by A4,A5,A7,A9,TOPREAL1:def 3;
        hence thesis by TARSKI:def 2;
      end;
      suppose
        j=2;
        then x = LSeg(p2,p3) by A2,A5,A3,A7,TOPREAL1:def 3;
        hence thesis by TARSKI:def 2;
      end;
    end;
    let x be object such that
A10: x in {LSeg(p1,p2),LSeg(p2,p3)};
    per cases by A10,TARSKI:def 2;
    suppose
A11:  x = LSeg(p1,p2);
A12:  1+1 <= len f by A2;
      x=LSeg(f,1) by A2,A4,A5,A6,A11,TOPREAL1:def 3;
      hence thesis by A12;
    end;
    suppose
      x = LSeg(p2,p3);
      then x=LSeg(f,2) by A2,A5,A3,TOPREAL1:def 3;
      hence thesis by A2;
    end;
  end;
  hence thesis by ZFMISC_1:75;
end;
