reserve i,j,k,n for Nat;
reserve D for non empty set,
  p for Element of D,
  f,g for FinSequence of D;

theorem Th52:
  p in rng f & j+1 in dom(f:-p) implies (f:-p)/.(j+1) = f/.(j+p..f )
proof
  assume that
A1: p in rng f and
A2: j+1 in dom(f:-p);
A3: j+p..f in dom f by A1,A2,Th51;
  set i = j+1;
A4: p..f <= len f by A1,FINSEQ_4:21;
  consider k being Element of NAT such that
A5: k+1 = p..f and
A6: f:-p = f/^k by A1,Th49;
  k <= p..f by A5,NAT_1:11;
  then
A7: k <= len f by A4,XXREAL_0:2;
  thus (f:-p)/.i = (f:-p).i by A2,PARTFUN1:def 6
    .= f.(i+k) by A2,A6,A7,RFINSEQ:def 1
    .= f/.(j+p..f) by A5,A3,PARTFUN1:def 6;
end;
