theorem Th37:
  x in rng p implies not x in rng(p -| x)
proof
  assume that
A1: x in rng p and
A2: x in rng(p -| x);
  reconsider n = x..p - 1 as Element of NAT by A1,Th22;
  set r = p | Seg n;
A3: r = p -| x by A1,Th33;
  then consider y being object such that
A4: y in dom r and
A5: r.y = x by A2,FUNCT_1:def 3;
A6: dom r = Seg n by A1,A3,Th35;
  then reconsider y as Element of NAT by A4;
  y <= n by A4,A6,FINSEQ_1:1;
  then
A7: y + 1 <= x..p by XREAL_1:19;
  y < y + 1 by XREAL_1:29;
  then dom r c= dom p & y < x..p by A7,RELAT_1:60,XXREAL_0:2;
  then p.y <> x by A4,Th24;
  hence thesis by A4,A5,FUNCT_1:47;
end;
