 reserve Rseq, Rseq1, Rseq2 for Function of [:NAT,NAT:],REAL;

theorem SH3:
for X being non empty set, s being sequence of X, n,m being Nat
  st m+1 in dom Shift(s|Segm n, 1) holds Shift(s|Segm n,1).(m+1) = s.m
proof
   let X be non empty set;
   let s be sequence of X;
   let n,m be Nat;
   assume m+1 in dom Shift(s|Segm n,1); then
   consider k be Nat such that
A1: k in dom(s|Segm n) & m+1 = k+1 by VALUED_1:39;
   Shift(s|Segm n,1).(m+1) = (s|Segm n).m by A1,VALUED_1:def 12;
   hence Shift(s|Segm n,1).(m+1) = s.m by A1,FUNCT_1:47;
end;
