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 Th42:
  for f be FinSequence of TOP-REAL 2 for p be Point of TOP-REAL 2
  st p in L~f holds L~L_Cut(f,p) c= L~f
proof
  let f be FinSequence of TOP-REAL 2;
  let p be Point of TOP-REAL 2 such that
A1: p in L~f;
  Index(p,f)<len f by A1,Th8;
  then
A2: Index(p,f)+1<=len f by NAT_1:13;
A3: 1<=Index(p,f) by A1,Th8;
  then
A4: 1<Index(p,f)+1 by NAT_1:13;
  then
A5: Index(p,f)+1 in dom f by A2,FINSEQ_3:25;
  len f <> 0 by A1,TOPREAL1:22;
  then
A6: len f >= 0+1 by NAT_1:13;
  then
A7: len f in dom f by FINSEQ_3:25;
  per cases;
  suppose
    p = f.(Index(p,f)+1);
    then L_Cut(f,p) = mid(f,Index(p,f)+1,len f) by Def3;
    hence thesis by A6,A4,A2,Lm7;
  end;
  suppose
    p <> f.(Index(p,f)+1);
    then
A8: L_Cut(f,p) = <*p*>^mid(f,Index(p,f)+1,len f) by Def3;
A9: f/.(Index(p,f)+1) in LSeg(f/.(Index(p,f)),f/.(Index(p,f)+1)) by RLTOPSP1:68
;
    p in LSeg(f,Index(p,f)) by A1,Th9;
    then
A10: p in LSeg(f/.(Index(p,f)),f/.(Index(p,f)+1)) by A3,A2,TOPREAL1:def 3;
A11: LSeg(f/.(Index(p,f)),f/.(Index(p,f)+1)) c= L~f by A1,A2,Th8,SPPOL_2:16;
    mid(f,Index(p,f)+1,len f)/.1 = f/.(Index(p,f)+1) by A7,A5,Lm10;
    then LSeg(p,mid(f,Index(p,f)+1,len f)/.1) c= LSeg(f/.(Index(p,f)),f/.(
    Index(p,f)+1)) by A10,A9,TOPREAL1:6;
    then
A12: LSeg(p,mid(f,Index(p,f)+1,len f)/.1) c= L~f by A11;
    mid(f,Index(p,f)+1,len f) <> {} by A7,A5,Lm8,CARD_1:27;
    then
A13: L~(<*p*>^mid(f,Index(p,f)+1,len f)) = LSeg(p,mid(f,Index(p,f)+1,len f
    )/.1) \/ L~mid(f,Index(p,f)+1,len f) by SPPOL_2:20;
    L~mid(f,Index(p,f)+1,len f) c= L~f by A6,A4,A2,Lm7;
    hence thesis by A8,A13,A12,XBOOLE_1:8;
  end;
end;
