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 Th44:
  f <> {} & f/.1 = p implies f-:p = <*p*> & f:-p = f
proof
  assume that
A1: f <> {} and
A2: f/.1 = p;
A3: p in rng f by A1,A2,Th42;
  p..f = 1 by A1,A2,Th43;
  then
A4: f -| p = {} by A3,FINSEQ_4:40;
  hence f-:p = {}^<*p*> by A3,Th40
    .= <*p*> by FINSEQ_1:34;
  thus f = {} ^ <* p *> ^ (f |-- p) by A3,A4,FINSEQ_4:51
    .= <* p *> ^ (f |-- p) by FINSEQ_1:34
    .= f:-p by A3,Th41;
end;
