reserve x,y for set;
reserve s,r for Real;
reserve r1,r2 for Real;
reserve n for Nat;
reserve p,q,q1,q2 for Point of TOP-REAL 2;

theorem Th49:
  for P1 being Subset of TOP-REAL 2,
  r being Real,p1,p2 being Point of TOP-REAL 2 st
  p1`1<=r & r<=p2`1 & P1 is_an_arc_of p1,p2 holds P1 meets Vertical_Line(r) &
  P1 /\ Vertical_Line(r) is closed
proof
  let P1 be Subset of TOP-REAL 2, r be Real,
  p1,p2 be Point of TOP-REAL 2;
  assume that
A1: p1`1<=r and
A2: r<=p2`1 and
A3: P1 is_an_arc_of p1,p2;
  reconsider P19 = P1 as non empty Subset of TOP-REAL 2 by A3,TOPREAL1:1;
  consider f being Function of I[01], (TOP-REAL 2)|P19 such that
A4: f is being_homeomorphism and
A5: f.0 = p1 and
A6: f.1 = p2 by A3,TOPREAL1:def 1;
A7: [#]((TOP-REAL 2)|P1)=P1 by PRE_TOPC:def 5;
  then reconsider f1=f as Function of the carrier of I[01],
  the carrier of TOP-REAL 2 by FUNCT_2:7;
  reconsider f2=f1 as Function of I[01],TOP-REAL 2;
  reconsider proj11=proj1 as Function of the carrier of TOP-REAL 2,REAL;
  reconsider proj12=proj1 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
  reconsider g1=proj11*f1 as Function of the carrier of I[01],REAL;
  reconsider g=g1 as Function of I[01],R^1 by TOPMETR:17;
  f is continuous by A4,TOPS_2:def 5;
  then
A8: f2 is continuous by Th3;
A9: proj12 is continuous by TOPREAL5:10;
A10: dom f=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
  then
A11: 0 in dom f by XXREAL_1:1;
A12: 1 in dom f by A10,XXREAL_1:1;
A13: g.0=proj1.p1 by A5,A11,FUNCT_1:13
    .=p1`1 by PSCOMP_1:def 5;
A14: g.1=proj1.p2 by A6,A12,FUNCT_1:13
    .=p2`1 by PSCOMP_1:def 5;
  reconsider P19 as non empty Subset of TOP-REAL 2;
A15: P19 is closed by A3,COMPTS_1:7,JORDAN5A:1;
A16: Vertical_Line(r) is closed by Th6;
  now per cases by A1,A2,A13,A14,BORSUK_1:def 14,def 15,XXREAL_0:1;
    case g.0=g.1;
      then
A17:  g.0=r by A1,A2,A13,A14,XXREAL_0:1;
A18:  f.0 in rng f by A11,FUNCT_1:def 3;
      then f.0 in P1 by A7;
      then reconsider p=f.0 as Point of TOP-REAL 2;
      p`1=proj1.(f.0) by PSCOMP_1:def 5
        .=r by A11,A17,FUNCT_1:13;
      then f.0 in {q where q is Point of TOP-REAL 2: q`1=r};
      hence P1 meets Vertical_Line(r) by A7,A18,XBOOLE_0:3;
    end;
    case
A19:  g.0[01]=r;
A20:  f.0 in rng f by A11,FUNCT_1:def 3;
      then f.0 in P19 by A7;
      then reconsider p=f.0 as Point of TOP-REAL 2;
      p`1=proj1.(f.0) by PSCOMP_1:def 5
        .=r by A11,A19,BORSUK_1:def 14,FUNCT_1:13;
      then f.0 in {q where q is Point of TOP-REAL 2: q`1=r};
      hence P1 meets Vertical_Line(r) by A7,A20,XBOOLE_0:3;
    end;
    case
A21:  g.1[01]=r;
A22:  f.1 in rng f by A12,FUNCT_1:def 3;
      then f.1 in P1 by A7;
      then reconsider p=f.1 as Point of TOP-REAL 2;
      p`1=proj1.(f.1) by PSCOMP_1:def 5
        .=r by A12,A21,BORSUK_1:def 15,FUNCT_1:13;
      then f.1 in {q where q is Point of TOP-REAL 2: q`1=r};
      hence P1 meets Vertical_Line(r) by A7,A22,XBOOLE_0:3;
    end;
    case g.0[01]< r & r < g.1[01];
      then consider r1 such that
A23:  0<r1 and
A24:  r1<1 and
A25:  g.r1=r by A8,A9,Lm1;
      dom f=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
      then
A26:  r1 in dom f by A23,A24,XXREAL_1:1;
A27:  [#]((TOP-REAL 2)|P1)=P1 by PRE_TOPC:def 5;
A28:  f.r1 in rng f by A26,FUNCT_1:def 3;
      then f.r1 in P19 by A27;
      then reconsider p=f.r1 as Point of TOP-REAL 2;
      p`1=proj1.(f.r1) by PSCOMP_1:def 5
        .=r by A25,A26,FUNCT_1:13;
      then f.r1 in {q where q is Point of TOP-REAL 2: q`1=r};
      hence P1 meets Vertical_Line(r) by A27,A28,XBOOLE_0:3;
    end;
  end;
  hence thesis by A15,A16,TOPS_1:8;
end;
