 reserve X,Y for set,
         n,m,k,i for Nat,
         r for Real,
         R for Element of F_Real,
         K for Field,
         f,f1,f2,g1,g2 for FinSequence,
         rf,rf1,rf2 for real-valued FinSequence,
         cf,cf1,cf2 for complex-valued FinSequence,
         F for Function;

theorem Th11:
  for f be FinSubsequence for n be Element of NAT st
     for i holds i+n in X iff i in Y
  holds (n Shift f) |X = n Shift(f|Y)
proof
  let f be FinSubsequence;
  let n be Element of NAT such that
   A1: i+n in X iff i in Y;
  set fY=f|Y;
  set nf=n Shift f;
  set nfX=nf|X;
  set nfY=n Shift fY;
  A2: dom nfY={k+n where k is Nat:k in dom fY} by VALUED_1:def 12;
  A3: now let x be object;
   assume x in dom nfY;
   then consider k be Nat such that
    A4: x=k+n and
    A5: k in dom fY by A2;
   A6: k in dom f by A5,RELAT_1:57;
   A7: k in Y by A5,RELAT_1:57;
   then x in X by A1,A4;
   hence nfX.x=nf.x by FUNCT_1:49
    .=f.k by A4,A6,VALUED_1:def 12
    .=fY.k by A7,FUNCT_1:49
    .=nfY.x by A4,A5,VALUED_1:def 12;
  end;
  A8: dom nf={k+n where k is Nat:k in dom f} by VALUED_1:def 12;
  A9: dom nfY c=dom nfX
  proof
   let x be object;
   assume x in dom nfY;
   then consider k be Nat such that
    A10: x=k+n and
    A11: k in dom fY by A2;
   k in Y by A11,RELAT_1:57;
   then A12: x in X by A1,A10;
   k in dom f by A11,RELAT_1:57;
   then x in dom nf by A8,A10;
   hence thesis by A12,RELAT_1:57;
  end;
  dom nfX c=dom nfY
  proof
   let x be object;
   assume A13: x in dom nfX;
   then x in dom nf by RELAT_1:57;
   then consider k be Nat such that
    A14: x=k+n and
    A15: k in dom f by A8;
   x in X by A13,RELAT_1:57;
   then k in Y by A1,A14;
   then k in dom fY by A15,RELAT_1:57;
   hence thesis by A2,A14;
  end;
  then dom nfY=dom nfX by A9;
  hence thesis by A3,FUNCT_1:2;
end;
