
theorem Th9:
  ex f be LinearOperator of RNS_Real,REAL-NS 1
  st f is isomorphism
   & for x be Element of RNS_Real
     holds f.x = <*x*>
  proof
    deffunc H1(Real) = In(<*$1*>, REAL 1);

    consider f be Function of REAL,REAL 1 such that
    A1: for x be Element of REAL
        holds f.x = H1(x) from FUNCT_2:sch 4;

    A2: for x be Element of RNS_Real
        holds f.x = <*x*>
    proof
      let x be Element of RNS_Real;
      reconsider x0 = x as Element of REAL;
      f.x = In(<*x*>, REAL 1) by A1;
      hence f.x = <*x*> by SUBSET_1:def 8;
    end;

    reconsider f as Function of
      the carrier of RNS_Real, the carrier of REAL-NS 1
      by REAL_NS1:def 4;

    for v,w be Element of RNS_Real
    holds f.(v + w) = f.v + f. w
    proof
      let v,w be Element of RNS_Real;
      reconsider v0 = v, w0 = w as Element of REAL;
      A4: f.v = <*v0*> by A2;
      A5: f.w = <*w0*> by A2;

      v + w = v0 + w0 by BINOP_2:def 9;
      hence f.(v + w)
       = <*v0 + w0*> by A2
      .= <*v0*> + <*w0*> by RVSUM_1:13
      .= f.v + f.w by A4,A5,REAL_NS1:2;
    end;
    then
    A7: f is additive;

    for x be VECTOR of RNS_Real
    for r be Real
    holds f.(r * x) = r * (f.x)
    proof
      let v be VECTOR of RNS_Real,
          r be Real;
      reconsider v0 = v as Element of REAL;
      r * v = r * v0 by BINOP_2:def 11;
      hence f.(r * v)
       = <* r * v0 *> by A2
      .= r * <*v0*> by RVSUM_1:47
      .= r * (f.v) by A2,REAL_NS1:3;
    end;
    then reconsider f as LinearOperator of RNS_Real,REAL-NS 1
      by A7,LOPBAN_1:def 5;

    take f;
    now
      let z1,z2 be object;
      assume
      A9: z1 in the carrier of RNS_Real
        & z2 in the carrier of RNS_Real
        & f.z1 = f.z2;

      <*z1*> = f.z1 by A2,A9
      .= <*z2*> by A2,A9;
      hence z1 = z2 by FINSEQ_1:76;
    end;
    then
    A10: f is one-to-one by FUNCT_2:19;

    now
      let w be object;
      assume w in the carrier of REAL-NS 1;
      then w in REAL 1 by REAL_NS1:def 4;
      then w in {s where s is Element of REAL* : len s = 1}
        by FINSEQ_2:def 4;
      then consider s be Element of REAL* such that
      A11: w = s & len s = 1;

      set x = s.1;
      A12: 1 in {1} by TARSKI:def 1;
      1 in dom s by A11,A12,FINSEQ_1:2,def 3;
      then s.1 in rng s by FUNCT_1:3;
      then
      reconsider x as Element of REAL;
      reconsider x as Element of RNS_Real;

      A13: w = <*x*> by A11,FINSEQ_1:40;
      f.x = w by A2,A13;
      hence w in rng f by FUNCT_2:112;
    end;
    then the carrier of REAL-NS 1 c= rng f;
    then
    A14: f is onto by FUNCT_2:def 3,XBOOLE_0:def 10;

    for x be Point of RNS_Real holds ||.x.|| = ||.f.x.||
    proof
      let x be Point of RNS_Real;
      reconsider x0 = x as Element of REAL;
      A15: <*x0*> in REAL 1;
      then reconsider x1 = <*x0*> as VECTOR of REAL-NS 1 by REAL_NS1:def 4;
      ||.f.x.||
       = |. <*x0*> .| by A2,REAL_NS1:1
      .= ||.x1.|| by A15,REAL_NS1:1
      .= |.x0.| by PDIFF_8:2
      .= absreal.x0 by EUCLID:def 2
      .= ||.x.||;
      hence thesis;
    end;
    hence thesis by A2,A10,A14,NORMSP_3:def 11;
  end;
