reserve P for Subset of TOP-REAL 2,
  f,f1,f2,g for FinSequence of TOP-REAL 2,
  p,p1,p2,q,q1,q2 for Point of TOP-REAL 2,
  r1,r2,r19,r29 for Real,
  i,j,k,n for Nat;

theorem Th20:
  f is non empty implies L~(<*p*>^f) = LSeg(p,f/.1) \/ L~f
proof
  set q = f/.1;
A1: len <*p*> = 1 by FINSEQ_1:39;
  then
A2: len (<*p*>^f) = 1 + len f by FINSEQ_1:22;
  assume that
A3: f is non empty;
  hereby
    let x be object;
    assume
A4: x in L~(<*p*>^f);
    then reconsider r = x as Point of TOP-REAL 2;
    consider i such that
A5: 1 <= i and
A6: i+1 <= len (<*p*>^f) and
A7: r in LSeg((<*p*>^f)/.i,(<*p*>^f)/.(i+1)) by A4,Th14;
    now
      per cases by A5,XXREAL_0:1;
      case
A8:     i = 1;
        then
A9:     p = (<*p*>^f)/.i by FINSEQ_5:15;
        i in dom f by A3,A8,FINSEQ_5:6;
        hence r in LSeg(p,q) by A1,A7,A8,A9,FINSEQ_4:69;
      end;
      case
A10:    i > 1;
        then consider j be Nat such that
A11:    i = j + 1 by NAT_1:6;
        reconsider j as Element of NAT by ORDINAL1:def 12;
A12:    1 <= j by A10,A11,NAT_1:13;
A13:    j+1 <= len f by A2,A6,A11,XREAL_1:6;
        then j+1 in dom f by A12,SEQ_4:134;
        then
A14:    (<*p*>^f)/.(i+1) = f/.(j+1) by A1,A11,FINSEQ_4:69;
        j in dom f by A12,A13,SEQ_4:134;
        then (<*p*>^f)/.i = f/.j by A1,A11,FINSEQ_4:69;
        hence r in L~f by A7,A12,A13,A14,Th15;
      end;
    end;
    hence x in LSeg(p,q) \/ L~f by XBOOLE_0:def 3;
  end;
  let x be object;
  assume
A15: x in LSeg(p,q) \/ L~f;
  then reconsider r = x as Point of TOP-REAL 2;
  per cases by A15,XBOOLE_0:def 3;
  suppose
A16: r in LSeg(p,q);
    set i = 1;
    i <= len f by A3,NAT_1:14;
    then
A17: i+1 <= len(<*p*>^f) by A2,XREAL_1:6;
    i in dom f by A3,FINSEQ_5:6;
    then
A18: q = (<*p*>^f)/.(i+1) by A1,FINSEQ_4:69;
    p = (<*p*>^f)/.i by FINSEQ_5:15;
    hence thesis by A16,A17,A18,Th15;
  end;
  suppose
    r in L~f;
    then consider j such that
A19: 1 <= j and
A20: j+1 <= len f and
A21: r in LSeg(f/.j,f/.(j+1)) by Th14;
    set i = j + 1;
    j in dom f by A19,A20,SEQ_4:134;
    then
A22: (<*p*>^f)/.i = f/.j by A1,FINSEQ_4:69;
    j+1 in dom f by A19,A20,SEQ_4:134;
    then
A23: (<*p*>^f)/.(i+1) = f/.(j+1) by A1,FINSEQ_4:69;
    i+1 <= len (<*p*>^f) by A2,A20,XREAL_1:6;
    hence thesis by A21,A22,A23,Th15,NAT_1:12;
  end;
end;
