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 Th41:
  for c being Curve of T,
      f being FinSequence of Curves T holds Sum(f ^ <*c*>) = Sum f + c
  proof
    let c be Curve of T;
    let f be FinSequence of Curves T;
    per cases;
    suppose
A1:   len f <= 0;
A2:   f = {} by A1;
      reconsider c0 = {} as Curve of T by Th21;
      thus Sum(f ^ <*c*>) = Sum <*c*> by A2,FINSEQ_1:34
      .= c0 \/ c by Th40
      .= c0 + c by Def12
      .= Sum f + c by Def14,A1;
    end;
    suppose
A3:   len f > 0;
      set f1 = f ^ <*c*>;
A4:   len f1 = len f + len <*c*> by FINSEQ_1:22
      .= len f + 1 by FINSEQ_1:39;
A5:   Sum f1 = (Partial_Sums f1).len f1 by Def14;
      0 < 0 + len f by A3;
      then
A6:  1 <= len f by NAT_1:19;
A7:  len f < len f1 by A4,NAT_1:13;
      len f in Seg len f1 by A6,A7,FINSEQ_1:1;
      then len f in Seg len Partial_Sums f1 by Def13;
      then len f in dom Partial_Sums f1 by FINSEQ_1:def 3;
      then
A8:   Partial_Sums(f1)/.(len f) = Partial_Sums(f1).(len f) by PARTFUN1:def 6
      .= (Partial_Sums f).(len f) by Lm2
      .= Sum f by A3,Def14;
      len f1 in Seg len f1 by FINSEQ_1:3;
      then len f1 in dom f1 by FINSEQ_1:def 3;
      then
A9:   f1/.(len f +1) = f1.(len f + 1) by A4,PARTFUN1:def 6
      .= c by FINSEQ_1:42;
      thus Sum(f ^ <*c*>) = Sum f + c by A8,A9,A5,A7,A4,A6,Def13;
    end;
  end;
