reserve i,j,k,n for Nat,
  X,Y,a,b,c,x for set,
  r,s for Real;

theorem Th5:
  for g1,g2 be FinSequence of TOP-REAL 2 holds L~g1 c= L~(g1^'g2)
proof
  let g1,g2 be FinSequence of TOP-REAL 2;
  let x be object;
  assume x in L~g1;
  then consider a such that
A1: x in a & a in { LSeg(g1,i) where i is Nat
     : 1 <= i & i+1 <= len g1 } by TARSKI:def 4;
  consider j being Nat such that
A2: a = LSeg(g1,j) and
A3: 1 <= j and
A4: j+1 <= len g1 by A1;
  j < len g1 by A4,NAT_1:13;
  then
A5: a = LSeg(g1^'g2,j) by A2,TOPREAL8:28;
  len g1 <= len (g1^'g2) by TOPREAL8:7;
  then j+1 <= len (g1^'g2) by A4,XXREAL_0:2;
  then a in { LSeg(g1^'g2,i)where i is Nat
     : 1 <= i & i+1 <= len (g1^'g2) } by A3,A5;
  hence thesis by A1,TARSKI:def 4;
end;
