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 Th26:
  R is well-ordering implies R-Seg(a),R-Seg(b) are_c=-comparable
proof
  assume
A1: R is well-ordering;
A2: now
    assume
A3: a in field R & b in field R;
    now
      assume a <> b;
A4:   now
        assume
A5:     [b,a] in R;
        now
          let c be object;
          assume
A6:       c in R-Seg(b);
          then
A7:       [c,b] in R by Th1;
          then
A8:       [c,a] in R by A1,A5,Lm2;
          c <> b by A6,Th1;
          then c <> a by A1,A5,A7,Lm3;
          hence c in R-Seg(a) by A8,Th1;
        end;
        hence R-Seg(b) c= R-Seg(a);
      end;
      now
        assume
A9:     [a,b] in R;
        now
          let c be object;
          assume
A10:      c in R-Seg(a);
          then
A11:      [c,a] in R by Th1;
          then
A12:      [c,b] in R by A1,A9,Lm2;
          c <> a by A10,Th1;
          then c <> b by A1,A9,A11,Lm3;
          hence c in R-Seg(b) by A12,Th1;
        end;
        hence R-Seg(a) c= R-Seg(b);
      end;
      hence thesis by A1,A3,A4,Lm4;
    end;
    hence thesis;
  end;
  now
    assume R-Seg(a) = {} or R-Seg(b) = {};
    then R-Seg(a) c= R-Seg(b) or R-Seg(b) c= R-Seg(a);
    hence thesis;
  end;
  hence thesis by A2,Th2;
end;
