reserve T,U for non empty TopSpace;
reserve t for Point of T;
reserve n for Nat;
reserve T for TopStruct;
reserve f for PartFunc of R^1, T;
reserve c for Curve of T;
reserve T for non empty TopStruct;

theorem Th42:
  for X being set, f being FinSequence of Curves T
  st (for i being Nat st 1 <= i & i <= len f holds rng(f/.i) c= X)
  holds rng Sum f c= X
  proof
    let X be set;
    defpred P[Nat] means
    for f being FinSequence of Curves T st len f = $1
    & (for i being Nat st 1 <= i & i <= len f holds rng(f/.i) c= X)
    holds rng Sum f c= X;
A1: P[0]
    proof
      let f be FinSequence of Curves T;
      assume len f = 0;
      then Sum f = {} by Def14;
      then rng Sum f = {};
      hence thesis;
    end;
A2: for k be Nat st P[k] holds P[k + 1]
    proof
      let k be Nat;
      assume
A3:   P[k];
      let f be FinSequence of Curves T;
      assume
A4:   len f = k+1;
      then consider f1 be FinSequence of Curves T,
               c be Element of Curves T such that
A5:   f = f1 ^ <*c*> by FINSEQ_2:19;
      assume
A6:   for i being Nat st 1 <= i & i <= len f holds rng(f/.i) c= X;
A7: len f = len f1 + len<*c*> by A5,FINSEQ_1:22
      .= len f1 + 1 by FINSEQ_1:39;
A8:  Sum f = Sum f1 + c by A5,Th41;
      per cases;
      suppose not Sum f1 \/ c is Curve of T;
        then Sum f = {} by A8,Def12;
        then rng Sum f = {};
        hence thesis;
      end;
      suppose Sum f1 \/ c is Curve of T;
        then
A9:     Sum f = Sum f1 \/ c by A8,Def12;
A10:     for i being Nat st 1 <= i & i <= len f1
        holds rng(f1/.i) c= X
        proof
          let i be Nat;
          assume
A11:       1 <= i & i <= len f1;
          then
A12:       i+1 <= len f1 + 1 by XREAL_1:6;
          i <= i+1 by NAT_1:12;
          then
A13:       i <= len f by A12,A7,XXREAL_0:2;
          then
A14:       rng(f/.i) c= X by A6,A11;
          i in Seg len f by A11,A13,FINSEQ_1:1;
          then i in dom f by FINSEQ_1:def 3;
          then
A15:       rng(f.i) c= X by A14,PARTFUN1:def 6;
          i in Seg len f1 by A11,FINSEQ_1:1;
          then
A16:       i in dom f1 by FINSEQ_1:def 3;
          then f.i = f1.i by A5,FINSEQ_1:def 7;
          hence thesis by A15,A16,PARTFUN1:def 6;
        end;
A17:     rng Sum f1 c= X by A3,A7,A4,A10;
        len f in Seg len f by A7,FINSEQ_1:3;
        then
A18:     len f in dom f by FINSEQ_1:def 3;
        f.(len f) = c by A7,A5,FINSEQ_1:42;
        then
A19:    f/.(len f) = c by A18,PARTFUN1:def 6;
        0 + 1 <= len f1 + 1 by XREAL_1:6;
        then
A20:     rng c c= X by A7,A19,A6;
        rng Sum f = rng(Sum f1) \/ rng c by A9,RELAT_1:12;
        hence thesis by A17,A20,XBOOLE_1:8;
      end;
    end;
A21: for k being Nat holds P[k] from NAT_1:sch 2(A1,A2);
    let f be FinSequence of Curves T;
    thus thesis by A21;
  end;
