reserve i,j,k,n,m for Nat;
reserve p,q for Point of TOP-REAL 2;
reserve G for Go-board;
reserve C for Subset of TOP-REAL 2;

theorem Th47:
  for f being rectangular FinSequence of TOP-REAL 2, p,q being
Point of TOP-REAL 2 st not q in L~f & <*p,q*> is_in_the_area_of f holds LSeg(p,
  q) /\ L~f c= {p}
proof
  let f be rectangular FinSequence of TOP-REAL 2, p,q be Point of TOP-REAL 2
  such that
A1: not q in L~f;
  assume
A2: <*p,q*> is_in_the_area_of f;
A3: dom <*p,q*> = {1,2} by FINSEQ_1:2,89;
  then
A4: 2 in dom <*p,q*> by TARSKI:def 2;
A5: <*p,q*>/.2 = q by FINSEQ_4:17;
  then
A6: W-bound L~f <= q`1 by A2,A4;
A7: <*q*> is_in_the_area_of f by A2,Th42;
  then q`1 <> W-bound L~f by A1,Th43;
  then
A8: W-bound L~f < q`1 by A6,XXREAL_0:1;
A9: q`2 <= N-bound L~f by A2,A4,A5;
  q`2 <> N-bound L~f by A1,A7,Th43;
  then
A10: q`2 < N-bound L~f by A9,XXREAL_0:1;
  let x be object;
A11: <*p,q*>/.1 = p by FINSEQ_4:17;
A12: q`1 <= E-bound L~f by A2,A4,A5;
  q`1 <> E-bound L~f by A1,A7,Th43;
  then
A13: q`1 < E-bound L~f by A12,XXREAL_0:1;
  assume
A14: x in LSeg(p,q) /\ L~f;
  then reconsider p9 = x as Point of TOP-REAL 2;
A15: p9 in L~f by A14,XBOOLE_0:def 4;
A16: 1 in dom <*p,q*> by A3,TARSKI:def 2;
  then
A17: W-bound L~f <= p`1 by A2,A11;
A18: p`2 <= N-bound L~f by A2,A16,A11;
A19: S-bound L~f <= p`2 by A2,A16,A11;
A20: p`1 <= E-bound L~f by A2,A16,A11;
A21: S-bound L~f <= q`2 by A2,A4,A5;
  q`2 <> S-bound L~f by A1,A7,Th43;
  then
A22: S-bound L~f < q`2 by A21,XXREAL_0:1;
  x in LSeg(p,q) by A14,XBOOLE_0:def 4;
  then consider r being Real such that
A23: p9 = (1-r)*p+r*q and
A24: 0<=r and
A25: r<=1;
A26: p9`1 = ((1-r)*p)`1+(r*q)`1 by A23,TOPREAL3:2
    .= (1-r)*p`1+(r*q)`1 by TOPREAL3:4
    .= (1-r)*p`1+r*q`1 by TOPREAL3:4;
A27: p9`2 = ((1-r)*p)`2+(r*q)`2 by A23,TOPREAL3:2
    .= (1-r)*p`2+(r*q)`2 by TOPREAL3:4
    .= (1-r)*p`2+r*q`2 by TOPREAL3:4;
  per cases by A15,Th32;
  suppose
    p9`1 = W-bound L~f;
    then r = 0 by A17,A8,A24,A25,A26,XREAL_1:180;
    then p9 = 1*p+0.TOP-REAL 2 by A23,RLVECT_1:10
      .= 1*p by RLVECT_1:4
      .= p by RLVECT_1:def 8;
    hence thesis by ZFMISC_1:31;
  end;
  suppose
    p9`1 = E-bound L~f;
    then r = 0 by A20,A13,A24,A25,A26,XREAL_1:179;
    then p9 = 1*p+0.TOP-REAL 2 by A23,RLVECT_1:10
      .= 1*p by RLVECT_1:4
      .= p by RLVECT_1:def 8;
    hence thesis by ZFMISC_1:31;
  end;
  suppose
    p9`2 = S-bound L~f;
    then r = 0 by A19,A22,A24,A25,A27,XREAL_1:180;
    then p9 = 1*p+0.TOP-REAL 2 by A23,RLVECT_1:10
      .= 1*p by RLVECT_1:4
      .= p by RLVECT_1:def 8;
    hence thesis by ZFMISC_1:31;
  end;
  suppose
    p9`2 = N-bound L~f;
    then r = 0 by A18,A10,A24,A25,A27,XREAL_1:179;
    then p9 = 1*p+0.TOP-REAL 2 by A23,RLVECT_1:10
      .= 1*p by RLVECT_1:4
      .= p by RLVECT_1:def 8;
    hence thesis by ZFMISC_1:31;
  end;
end;
