reserve i,j,k,m,n for Nat,
  D for non empty set,
  p for Element of D,
  f for FinSequence of D;
reserve D for non empty set,
  p for Element of D,
  f for FinSequence of D;
reserve f for circular FinSequence of D;
reserve f,g for FinSequence of TOP-REAL 2;

theorem Th22:
  rng f c= rng g implies rng Y_axis f c= rng Y_axis g
proof
  assume
A1: rng f c= rng g;
A2: dom Y_axis g = dom g by SPRECT_2:16;
  let x be object;
  assume x in rng Y_axis f;
  then consider y being object such that
A3: y in dom Y_axis f and
A4: (Y_axis f).y = x by FUNCT_1:def 3;
  reconsider y as Element of NAT by A3;
A5: (Y_axis f).y = (f/.y)`2 by A3,GOBOARD1:def 2;
  dom Y_axis f = dom f by SPRECT_2:16;
  then f/.y in rng f by A3,PARTFUN2:2;
  then consider z being object such that
A6: z in dom g and
A7: g.z = f/.y by A1,FUNCT_1:def 3;
  reconsider z as Element of NAT by A6;
  g/.z = f/.y by A6,A7,PARTFUN1:def 6;
  then (Y_axis g).z = (f/.y)`2 by A2,A6,GOBOARD1:def 2;
  hence thesis by A4,A2,A5,A6,FUNCT_1:def 3;
end;
