reserve a,b,c,d,x,y,z for object, X,Y,Z for set;
reserve R,S,T for Relation;
reserve F,G for Function;

theorem Th34:
  R is well-ordering & a in field R & b in field R & (for c st c
  in R-Seg(a) holds [c,b] in R & c <> b) implies [a,b] in R
proof
  assume that
A1: R is well-ordering & a in field R & b in field R and
A2: c in R-Seg(a) implies [c,b] in R & c <> b;
  assume
A3: not [a,b] in R;
  a <> b by A1,A3,Lm1;
  then [b,a] in R by A1,A3,Lm4;
  then b in R-Seg(a) by A3,Th1;
  hence contradiction by A2;
end;
