theorem Th38:
  for pai1,pai2 being inf_path of R, k being Element of NAT holds
  pai1.k = pai2.0 implies PathConc(pai1,pai2,k) is inf_path of R
proof
  let pai1,pai2 be inf_path of R;
  let k be Element of NAT;
  set pai = PathConc(pai1,pai2,k);
  assume
A1: pai1.k = pai2.0;
  for n being Nat holds [pai.n,pai.(n+1)] in R
  proof
    let n be Nat;
    set n1=n+1;
    per cases by XXREAL_0:1;
    suppose
A2:   n1 < k;
      then
A3:   n < k by NAT_1:13;
A4:   pai.n = PathChange(pai1,pai2,k,n) by Def69
        .= pai1.n by A3,Def68;
      pai.n1 = PathChange(pai1,pai2,k,n1) by Def69
        .=pai1.n1 by A2,Def68;
      hence thesis by A4,Def39;
    end;
    suppose
A5:   n1 = k;
      then
A6:   n < k by NAT_1:13;
A7:   pai.n = PathChange(pai1,pai2,k,n) by Def69
        .= pai1.n by A6,Def68;
      pai.n1 = PathChange(pai1,pai2,k,n1) by Def69
        .= pai2.(n1-k) by A5,Def68
        .= pai1.n1 by A1,A5;
      hence thesis by A7,Def39;
    end;
    suppose
A8:   k < n1;
      then
A9:   k <= n by NAT_1:13;
      then reconsider m = n - k as Element of NAT by NAT_1:21;
A10:  pai.n1 = PathChange(pai1,pai2,k,n1) by Def69
        .= pai2.(n1-k) by A8,Def68
        .= pai2.(m+1);
      pai.n = PathChange(pai1,pai2,k,n) by Def69
        .= pai2.m by A9,Def68;
      hence thesis by A10,Def39;
    end;
  end;
  hence thesis by Def39;
end;
