
theorem Th71:
  for p be Polynomial of F_Complex ex f be Function of COMPLEX,
  COMPLEX st f = Polynomial-Function(F_Complex,p) & f is_continuous_on COMPLEX
proof
  set FuFF=Funcs(COMPLEX,COMPLEX);
  let p be Polynomial of F_Complex;
  reconsider fzero = COMPLEX --> 0c as Element of FuFF by FUNCT_2:9;
  defpred P[Nat,set] means $2 = FPower(p.($1-'1),$1-'1);
A1: the carrier of F_Complex = COMPLEX by COMPLFLD:def 1;
  then reconsider f = Polynomial-Function(F_Complex,p) as Function of COMPLEX,
  COMPLEX;
  deffunc F(Element of FuFF,Element of FuFF) = $1+$2;
  take f;
  thus f = Polynomial-Function(F_Complex,p);
A2: for x,y being Element of FuFF holds F(x,y) in FuFF by FUNCT_2:8;
  consider fadd be BinOp of FuFF such that
A3: for x,y be Element of FuFF holds fadd.(x,y) = F(x,y) from FUNCT_7:
  sch 1(A2);
  reconsider L=addLoopStr(#FuFF,fadd,fzero#) as non empty addLoopStr;
A4: now
    let u,v,w be Element of L;
    reconsider u1=u, v1=v, w1=w as Function of COMPLEX,COMPLEX by FUNCT_2:66;
A5: u1+v1 in Funcs(COMPLEX,COMPLEX) by FUNCT_2:9;
A6: v1+w1 in Funcs(COMPLEX,COMPLEX) by FUNCT_2:9;
    thus (u+v)+w = fadd.(u1+v1,w) by A3
      .= u1+v1+w1 by A3,A5
      .= u1+(v1+w1) by CFUNCT_1:13
      .= fadd.(u,v1+w1) by A3,A6
      .= u+(v+w) by A3;
  end;
A7: now
    let v be Element of L;
    reconsider v1=v as Function of COMPLEX,COMPLEX by FUNCT_2:66;
A8: now
      let x be Element of COMPLEX;
      thus (v1+fzero).x = v1.x+fzero.x by VALUED_1:1
        .= v1.x+0c by FUNCOP_1:7
        .= v1.x;
    end;
    thus v + 0.L = v1+fzero by A3
      .= v by A8,FUNCT_2:63;
  end;
  L is right_complementable
  proof
    let v be Element of L;
    reconsider v1=v as Function of COMPLEX,COMPLEX by FUNCT_2:66;
    reconsider w=-v1 as Element of L by FUNCT_2:9;
    take w;
A9: now
      let x be Element of COMPLEX;
      thus (v1+-v1).x = v1.x+(-v1).x by VALUED_1:1
        .= v1.x+-v1.x by VALUED_1:8
        .= fzero.x by FUNCOP_1:7;
    end;
    thus v + w = v1+-v1 by A3
      .= 0.L by A9,FUNCT_2:63;
  end;
  then reconsider
  L as add-associative right_zeroed right_complementable non empty
  addLoopStr by A4,A7,RLVECT_1:def 3,def 4;
A10: now
    let n be Nat;
    reconsider x = FPower(p.(n-'1),n-'1) as Element of L by A1,FUNCT_2:9;
    assume n in Seg len p;
    take x;
    thus P[n,x];
  end;
  consider F be FinSequence of the carrier of L such that
A11: dom F = Seg len p and
A12: for n be Nat st n in Seg len p holds P[n,F.n] from FINSEQ_1:sch 5(
  A10 );
A13: F|len F = F by FINSEQ_1:58;
  reconsider SF = Sum F as Function of COMPLEX,COMPLEX by FUNCT_2:66;
A14: now
    let x be Element of COMPLEX;
    reconsider x1=x as Element of F_Complex by COMPLFLD:def 1;
    consider H be FinSequence of the carrier of F_Complex such that
A15: eval(p,x1) = Sum H and
A16: len H = len p and
A17: for n be Element of NAT st n in dom H holds H.n = p.(n-'1)*(power
    F_Complex).(x1,n-'1) by POLYNOM4:def 2;
    defpred P[Nat] means
for SFk be Function of COMPLEX,COMPLEX st
    SFk = Sum (F|$1) holds Sum (H|$1) = SFk.x;
A18: len F = len p by A11,FINSEQ_1:def 3;
A19: for k be Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
       reconsider kk=k as Element of NAT by ORDINAL1:def 12;
      assume
A20:  for SFk be Function of COMPLEX,COMPLEX st SFk = Sum (F|k) holds
      Sum (H|k) = SFk.x;
      reconsider SFk1 = Sum (F|k) as Function of COMPLEX,COMPLEX by FUNCT_2:66;
      let SFk be Function of COMPLEX,COMPLEX;
      assume
A21:  SFk = Sum (F|(k+1));
      per cases;
      suppose
A22:    len F > k;
        reconsider g2 = FPower(p.kk,k) as Function of COMPLEX,COMPLEX by A1;
A23:    k+1 >= 1 by NAT_1:11;
        k+1 <= len F by A22,NAT_1:13;
        then
A24:    k+1 in dom F by A23,FINSEQ_3:25;
        then
A25:    F/.(k+1) = F.(k+1) by PARTFUN1:def 6
          .= FPower(p.(k+1-'1),k+1-'1) by A11,A12,A24
          .= FPower(p.kk,k+1-'1) by NAT_D:34
          .= FPower(p.kk,k) by NAT_D:34;
        F|(k+1) = F|k ^ <*F.(k+1)*> by A22,FINSEQ_5:83
          .= F|k ^ <*F/.(k+1)*> by A24,PARTFUN1:def 6;
        then
A26:    SFk = Sum(F|k) + F/.(k+1) by A21,FVSUM_1:71
          .= SFk1+g2 by A3,A25;
A27:    Sum (H|k) = SFk1.x by A20;
A28:    dom F = dom H by A11,A16,FINSEQ_1:def 3;
        then
A29:    H/.(k+1) = H.(k+1) by A24,PARTFUN1:def 6
          .= p.(k+1-'1)*(power F_Complex).(x1,k+1-'1) by A17,A28,A24
          .= p.kk*(power F_Complex).(x1,k+1-'1) by NAT_D:34
          .= p.kk*power(x1,k) by NAT_D:34
          .= FPower(p.kk,k).x by Def12;
        H|(k+1) = H|k ^ <*H.(k+1)*> by A16,A18,A22,FINSEQ_5:83
          .= H|k ^ <*H/.(k+1)*> by A28,A24,PARTFUN1:def 6;
        hence Sum (H|(k+1)) = Sum(H|k) + H/.(k+1) by FVSUM_1:71
          .= SFk.x by A29,A26,A27,VALUED_1:1;
      end;
      suppose
A30:    len F <= k;
        k <= k+1 by NAT_1:11;
        then
A31:    F|(k+1) = F & H|(k+1) = H by A16,A18,A30,FINSEQ_1:58,XXREAL_0:2;
        F|k = F & H|k = H by A16,A18,A30,FINSEQ_1:58;
        hence thesis by A20,A21,A31;
      end;
    end;
A32: P[0]
    proof
      let SFk be Function of COMPLEX,COMPLEX;
A33:  F|0 = <*>the carrier of L;
      assume SFk = Sum (F|0);
      then
A34:  SFk = 0.L by A33,RLVECT_1:43
        .= COMPLEX --> 0c;
      H|0 = <*>the carrier of F_Complex;
      hence Sum (H|0) = 0.F_Complex by RLVECT_1:43
        .= SFk.x by A34,COMPLFLD:7,FUNCOP_1:7;
    end;
A35: for k be Nat holds P[k] from NAT_1:sch 2(A32,A19);
A36: Sum(F|len F) = SF by FINSEQ_1:58;
    thus f.x = Sum H by A15,Def13
      .= Sum (H|len H) by FINSEQ_1:58
      .= SF.x by A16,A18,A35,A36;
  end;
  defpred P[Nat] means
for g be PartFunc of COMPLEX,COMPLEX st g =
  Sum (F|$1) holds g is_continuous_on COMPLEX;
A37: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    reconsider kk=k as Element of NAT by ORDINAL1:def 12;
    reconsider g1 = Sum (F|k) as Function of COMPLEX,COMPLEX by FUNCT_2:66;
    assume
A38: for g be PartFunc of COMPLEX,COMPLEX st g = Sum (F|k) holds g
    is_continuous_on COMPLEX;
    then
A39: g1 is_continuous_on COMPLEX;
    let g be PartFunc of COMPLEX,COMPLEX;
    assume
A40: g = Sum (F|(k+1));
    per cases;
    suppose
A41:  len F > k;
A42:  k+1 >= 1 by NAT_1:11;
      k+1 <= len F by A41,NAT_1:13;
      then
A43:  k+1 in dom F by A42,FINSEQ_3:25;
      then
A44:  F/.(k+1) = F.(k+1) by PARTFUN1:def 6
        .= FPower(p.(k+1-'1),k+1-'1) by A11,A12,A43
        .= FPower(p.kk,k+1-'1) by NAT_D:34
        .= FPower(p.kk,k) by NAT_D:34;
      consider g2 be Function of COMPLEX,COMPLEX such that
A45:  g2 = FPower(p.kk,k) and
A46:  g2 is_continuous_on COMPLEX by Th70;
      F|(k+1) = F|k ^ <*F.(k+1)*> by A41,FINSEQ_5:83
        .= F|k ^ <*F/.(k+1)*> by A43,PARTFUN1:def 6;
      then g = Sum(F|k) + F/.(k+1) by A40,FVSUM_1:71
        .= g1+g2 by A3,A44,A45;
      hence thesis by A39,A46,CFCONT_1:43;
    end;
    suppose
A47:  len F <= k;
      k <= k+1 by NAT_1:11;
      then F|(k+1) = F by A47,FINSEQ_1:58,XXREAL_0:2
        .= F|k by A47,FINSEQ_1:58;
      hence thesis by A38,A40;
    end;
  end;
A48: P[0]
  proof
    let g be PartFunc of COMPLEX,COMPLEX;
A49: F|0 = <*>the carrier of L;
    assume g = Sum(F|0);
    then g = 0.L by A49,RLVECT_1:43
      .= COMPLEX --> 0c;
    hence thesis by Th63;
  end;
  for k be Nat holds P[k] from NAT_1:sch 2(A48,A37);
  hence thesis by A13,A14,FUNCT_2:63;
end;
