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 Th76:
  p in rng f implies (f-:p)^((f:-p)/^1) = f
proof
A1: rng f c= D by FINSEQ_1:def 4;
  assume
A2: p in rng f;
  then rng(f|--p) c= rng f by FINSEQ_4:44;
  then rng(f|--p) c= D by A1;
  then reconsider f1 = f|--p as FinSequence of D by FINSEQ_1:def 4;
  thus (f-:p)^((f:-p)/^1) = (f-|p)^<*p*>^((f:-p)/^1) by A2,Th40
    .= (f-|p)^<*p*>^((<*p*>^f1)/^1) by A2,Th41
    .= (f-|p)^<*p*>^(f|--p) by Th45
    .= f by A2,FINSEQ_4:51;
end;
