reserve f for Function;
reserve p,q for FinSequence;
reserve A,B,C for set,x,x1,x2,y,z for object;
reserve k,l,m,n for Nat;
reserve a for Nat;
reserve D for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve L,M for Element of NAT;

theorem
  x in rng p & p is one-to-one implies rng(p -| x) misses rng(p |-- x)
proof
  assume that
A1: x in rng p and
A2: p is one-to-one;
  p = (p -| x) ^ <* x *> ^ (p |-- x) by A1,Th51;
  then rng(p |-- x) misses rng((p -| x) ^ <* x *>) by A2,FINSEQ_3:91;
  hence thesis by FINSEQ_1:29,XBOOLE_1:63;
end;
