reserve r1,r2 for Real;
reserve n,i,i1,i2,j for Nat;
reserve D for non empty set;
reserve f for FinSequence of D;

theorem Th41:
  for f being FinSequence of TOP-REAL 2, p being Point of TOP-REAL
  2 st p in L~f holds L~R_Cut(f,p) c= L~f
proof
  let f be FinSequence of TOP-REAL 2, p be Point of TOP-REAL 2 such that
A1: p in L~f;
A2: 1<=Index(p,f) by A1,Th8;
  len f <> 0 by A1,TOPREAL1:22;
  then
A3: len f >= 0+1 by NAT_1:13;
A4: Index(p,f)<=len f by A1,Th8;
  per cases;
  suppose
    p = f.1;
    then R_Cut(f,p)=<*p*> by Def4;
    then len R_Cut(f,p)=1 by FINSEQ_1:39;
    then L~R_Cut(f,p) = {} by TOPREAL1:22;
    hence thesis;
  end;
  suppose
A5: p<>f.1;
A6: f/.(Index(p,f)) in LSeg(f/.(Index(p,f)),f/.(Index(p,f)+1)) by RLTOPSP1:68;
A7: len (mid(f,1,Index(p,f))) =Index(p,f)-'1+1 by A3,A2,A4,FINSEQ_6:118
      .=Index(p,f) by A1,Th8,XREAL_1:235;
    then (mid(f,1,Index(p,f)))<><*>(the carrier of TOP-REAL 2) by A2;
    then
A8: L~(mid(f,1,Index(p,f))^<*p*>) = L~mid(f,1,Index(p,f)) \/ LSeg((mid(f,
    1,Index(p,f))/.Index(p,f)),p) by A7,SPPOL_2:19;
    mid(f,1,Index(p,f)) =(f/^(1-'1))|(Index(p,f)-'1+1) by A2,FINSEQ_6:def 3
      .=(f/^(0))|(Index(p,f)-'1+1) by XREAL_1:232
      .=f|(Index(p,f)-'1+1)
      .=f|Index(p,f) by A1,Th8,XREAL_1:235;
    then
A9: L~mid(f,1,Index(p,f)) c= L~f by TOPREAL3:20;
    Index(p,f) < len f by A1,Th8;
    then
A10: Index(p,f)+1<=len f by NAT_1:13;
    then
A11: LSeg(f/.(Index(p,f)),f/.(Index(p,f)+1)) c= L~f by A1,Th8,SPPOL_2:16;
    p in LSeg(f,Index(p,f)) by A1,Th9;
    then
A12: p in LSeg(f/.(Index(p,f)),f/.(Index(p,f)+1)) by A2,A10,TOPREAL1:def 3;
    (mid(f,1,Index(p,f)))/.Index(p,f)=mid(f,1,Index(p,f)).Index(p,f) by A2,A7,
FINSEQ_4:15
      .=f.(Index(p,f)) by A2,A4,FINSEQ_6:123
      .=f/.(Index(p,f)) by A1,A4,Th8,FINSEQ_4:15;
    then
    LSeg((mid(f,1,Index(p,f))/.Index(p,f)),p) c= LSeg(f/.(Index(p,f)),f/.
    (Index(p,f)+1)) by A12,A6,TOPREAL1:6;
    then
A13: LSeg((mid(f,1,Index(p,f))/.Index(p,f)),p)c= L~f by A11;
    R_Cut(f,p)=mid(f,1,Index(p,f))^<*p*> by A5,Def4;
    hence thesis by A8,A13,A9,XBOOLE_1:8;
  end;
end;
