reserve x,y,z for set;
reserve f,f1,f2,f3 for FinSequence,
  p,p1,p2,p3 for set,
  i,k for Nat;
reserve D for non empty set,
  p,p1,p2,p3 for Element of D,
  f,f1,f2 for FinSequence of D;

theorem Th72:
  p in rng f implies p..(f-:p) = p..f
proof
  assume
A1: p in rng f;
  then
A2: p..f <= len(f-:p) by FINSEQ_5:42;
A3: now
    p..f <> 0 by A1,FINSEQ_4:21;
    then p..f in Seg(p..f) by FINSEQ_1:3;
    then
A4: (f-:p)/.(p..f) = f/.(p..f) by A1,FINSEQ_5:43;
    let i such that
A5: 1 <= i and
A6: i < p..f;
    i in Seg(p..f) by A5,A6;
    then
A7: (f-:p)/.i = f/.i by A1,FINSEQ_5:43;
    p..f <= len f by A1,FINSEQ_4:21;
    then i <= len f by A6,XXREAL_0:2;
    then
A8: i in dom f by A5,FINSEQ_3:25;
    f/.(p..f) = p by A1,FINSEQ_5:38;
    hence (f-:p)/.i <> (f-:p)/.(p..f) by A6,A7,A4,A8,FINSEQ_5:39;
  end;
  1 <= p..f by A1,FINSEQ_4:21;
  then
A9: p..f in dom(f-:p) by A2,FINSEQ_3:25;
  (f-:p)/.(p..f) = p by A1,FINSEQ_5:45;
  hence thesis by A9,A3,Th48;
end;
