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
  for f,g being FinSequence of TOP-REAL 2, p being Point of TOP-REAL 2
st f.len f=g.1 & p in L~g & f is being_S-Seq & g is being_S-Seq & L~f /\ L~g={g
  .1} & p<>g.1 holds mid(f,1,len f -'1)^R_Cut(g,p) is being_S-Seq
proof
  let f,g be FinSequence of TOP-REAL 2, p be Point of TOP-REAL 2;
  assume that
A1: f.len f=g.1 and
A2: p in L~g and
A3: f is being_S-Seq and
A4: g is being_S-Seq and
A5: L~f /\ L~g={g.1} and
A6: p<>g.1;
  mid(f,1,len f -'1)^R_Cut(g,p) is_S-Seq_joining f/.1,p by A1,A2,A3,A4,A5,A6
,Th47;
  hence thesis;
end;
