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

theorem Th20:
  for R being Relation, a,b being object holds
    R reduces a,b iff a = b or [a,b] in R[*]
proof
  let R be Relation, a,b be object;
  hereby
    assume
A1: R reduces a,b;
    then consider p being RedSequence of R such that
A2: a = p.1 & b = p.len p;
A3: now
      let i be Nat;
      assume i >= 1 & i < len p;
      then i in dom p & i+1 in dom p by Lm3,Lm4;
      hence [p.i,p.(i+1)] in R by Def2;
    end;
    assume a <> b;
    then
A4: a in field R & b in field R by A1,Th18;
    len p >= 0+1 by NAT_1:13;
    hence [a,b] in R[*] by A2,A4,A3,FINSEQ_1:def 17;
  end;
  assume that
A5: a = b or [a,b] in R[*] and
A6: not R reduces a,b;
  consider p being FinSequence such that
A7: len p >= 1 and
A8: p.1 = a & p.(len p) = b and
A9: for i being Nat st i >= 1 & i < len p holds [p.i,p.(i+1)] in R by A5,A6
,Th12,FINSEQ_1:def 17;
  p is RedSequence of R
  proof
    thus len p > 0 by A7;
    let i be Nat;
    assume that
A10: i in dom p and
A11: i+1 in dom p;
    i >= 1 by A10,Lm1;
    hence thesis by A9,A11,Lm2;
  end;
  hence contradiction by A6,A8;
end;
