reserve p,q,r for FinSequence,
  x,y for object;

theorem Th49:
  for R being Relation st R is subcommutative for a,b,c being object
  st R reduces a,b & [a,c] in R holds b,c are_convergent_wrt R
proof
  let R be Relation;
  assume
A1: R is subcommutative;
  let a,b,c be object;
  given p being RedSequence of R such that
A2: p.1 = a and
A3: p.len p = b;
  defpred P[Nat] means $1 in dom p implies ex d being object st ([p.$1
  ,d] in R or p.$1 = d) & R reduces c,d;
  assume
A4: [a,c] in R;
  now
    let i be Nat such that
A5: i in dom p implies P[i] and
A6: i+1 in dom p;
    per cases;
    suppose
A7:   i = 0;
      R reduces c,c by Th12;
      hence P[i+1] by A2,A4,A7;
    end;
    suppose
A8:   i > 0;
A9:   i < len p by A6,Lm2;
      then consider d being object such that
A10:  [p.i,d] in R or p.i = d and
A11:  R reduces c,d by A5,A8,Lm3;
      i in dom p by A8,A9,Lm3;
      then
A12:  [p.i,p.(i+1)] in R by A6,Def2;
A13:  now
        assume [p.i,d] in R;
        then p.(i+1),d are_convergent<=1_wrt R by A1,A12;
        then consider e being object such that
A14:    [p.(i+1),e] in R or p.(i+1) = e and
A15:    [d,e] in R or d = e;
        take e;
        thus [p.(i+1),e] in R or p.(i+1) = e by A14;
        R reduces d,e by A15,Th12,Th15;
        hence R reduces c,e by A11,Th16;
      end;
      now
        assume p.i = d;
        then R reduces d, p.(i+1) by A12,Th15;
        hence R reduces c, p.(i+1) by A11,Th16;
      end;
      hence P[i+1] by A10,A13;
    end;
  end;
  then
A16: for k be Nat st P[k] holds P[k+1];
A17: len p in dom p by FINSEQ_5:6;
A18: P[ 0 ] by Lm1;
  for i being Nat holds P[i] from NAT_1:sch 2(A18,A16);
  then consider d being object such that
A19: ( [b,d] in R or b = d)& R reduces c,d by A3,A17;
  take d;
  thus thesis by A19,Th12,Th15;
end;
