reserve m,j,p,q,n,l for Element of NAT;
reserve e1,e2 for ExtReal;
reserve i for Nat,
        k,k1,k2,j1 for Element of NAT,
        x,x1,x2,y for set;
reserve p1,p2 for FinSequence;

theorem Th51:
  for p,q being FinSubsequence st q c= p holds Shift(q,i) c= Shift(p,i)
proof
  let p,q be FinSubsequence;
  assume
A1: q c= p;
A2: dom Shift(q,i) = {k+i where k is Nat: k in dom q} by Def12;
A3: dom Shift(p,i) = {k+i where k is Nat: k in dom p} by Def12;
  let x,y be object;
  assume
A4: [x,y] in Shift(q,i);
  then
A5: x in dom Shift(q,i) by FUNCT_1:1;
A6: y = Shift(q,i).x by A4,FUNCT_1:1;
  consider k being Nat such that
A7: x = k+i and
A8: k in dom q by A2,A5;
A9: dom q c= dom p by A1,GRFUNC_1:2;
  then
A10: x in dom Shift(p,i) by A3,A7,A8;
  y = q.k by A6,A7,A8,Def12
    .= p.k by A1,A8,GRFUNC_1:2
    .= Shift(p,i).x by A7,A8,A9,Def12;
  hence thesis by A10,FUNCT_1:1;
end;
