reserve P for Subset of TOP-REAL 2,
  f,f1,f2,g for FinSequence of TOP-REAL 2,
  p,p1,p2,q,q1,q2 for Point of TOP-REAL 2,
  r1,r2,r19,r29 for Real,
  i,j,k,n for Nat;

theorem Th51:
  p1,p2 split P & q in P & q <> p1 implies p1,q split P
proof
  assume p1 <> p2;
  given f1,f2 being S-Sequence_in_R2 such that
A1: p1 = f1/.1 and
A2: p1 = f2/.1 and
A3: p2 = f1/.len f1 and
A4: p2 = f2/.len f2 and
A5: L~f1 /\ L~f2 = {p1,p2} and
A6: P = L~f1 \/ L~f2;
  assume
A7: q in P;
  assume
A8: q <> p1;
  hence p1 <> q;
  per cases by A6,A7,XBOOLE_0:def 3;
  suppose
A9: q in L~f1;
    now
      per cases;
      suppose
        q in rng f1;
        hence thesis by A1,A2,A3,A4,A5,A6,A8,Lm16;
      end;
      suppose
A10:    not q in rng f1;
        consider i such that
A11:    1 <= i and
A12:    i+1 <= len f1 and
A13:    q in LSeg(f1,i) by A9,Th13;
        reconsider f19 = Ins(f1,i,q) as S-Sequence_in_R2 by A10,A13,Th48;
A14:    L~f19 = L~f1 by A13,Th25;
        1 <= i + 1 by NAT_1:11; then
        1 <= len f1 by A12,XXREAL_0:2; then
Z:      1 in dom f1 & len f1 in dom f1 by FINSEQ_3:25; then
a3:     p2 = f1.len f1 by A3,PARTFUN1:def 6;   
S2:     len f19 = len f1 + 1 by FINSEQ_5:69;
        1 <= len f1 + 1 by NAT_1:11; then
        1 <= len f19 by S2; then
S1:     len f19 in dom f19 & 1 in dom f19 by FINSEQ_3:25;
A15:    p2 = f19.len f19 by A12,a3,FINSEQ_5:74,S2 
            .= f19/.len f19 by S1,PARTFUN1:def 6;
A16:    q in rng f19 by FINSEQ_5:71;
        p1 = f1.1 by Z,A1,PARTFUN1:def 6; then
        p1 = f19.1 by A11,FINSEQ_5:75 .= f19/.1 by PARTFUN1:def 6,S1;
        hence thesis by A2,A4,A5,A6,A8,A16,A15,A14,Lm16;
      end;
    end;
    hence thesis;
  end;
  suppose
A17: q in L~f2;
    now
      per cases;
      suppose
        q in rng f2;
        hence thesis by A1,A2,A3,A4,A5,A6,A8,Lm16;
      end;
      suppose
A18:    not q in rng f2;
        consider i such that
A19:    1 <= i and
A20:    i+1 <= len f2 and
A21:    q in LSeg(f2,i) by A17,Th13;
        reconsider f29 = Ins(f2,i,q) as S-Sequence_in_R2 by A18,A21,Th48;
A22:    L~f29 = L~f2 by A21,Th25;
        1 <= i + 1 by NAT_1:11; then
        1 <= len f2 by A20,XXREAL_0:2; then
Z:      1 in dom f2 & len f2 in dom f2 by FINSEQ_3:25; then
Sa:     p2 = f2.len f2 by PARTFUN1:def 6,A4;
Sc:     len f29 = len f2 + 1 by FINSEQ_5:69;
        then
        1 <= len f29 by NAT_1:11; then
Sd:     1 in dom f29 & len f29 in dom f29 by FINSEQ_3:25;
A23:    p2 = f29.len f29 by Sc,Sa,A20,FINSEQ_5:74
           .= f29/.len f29 by PARTFUN1:def 6,Sd;
A24:    q in rng f29 by FINSEQ_5:71;
        p1 = f2.1 by PARTFUN1:def 6,A2,Z; then
        p1 = f29.1 by A19,FINSEQ_5:75 .= f29/.1 by Sd,PARTFUN1:def 6;
        hence thesis by A1,A3,A5,A6,A8,A24,A23,A22,Lm16;
      end;
    end;
    hence thesis;
  end;
end;
