
theorem Th13:
  for L being RelStr, C being set, R being auxiliary(i) (Relation
of L) st R satisfies_SIC_on C holds for x, z being Element of L holds x in C &
z in C & [x,z] in R & x <> z implies ex y being Element of L st y in C & [x,y]
  in R & [y,z] in R & x < y
proof
  let L be RelStr, C be set, R be auxiliary(i) Relation of L such that
A1: R satisfies_SIC_on C;
  let x, z be Element of L;
  assume that
A2: x in C and
A3: z in C and
A4: [x,z] in R and
A5: x <> z;
  consider y being Element of L such that
A6: y in C and
A7: [x,y] in R and
A8: [y,z] in R and
A9: x <> y by A2,A3,A4,A5,A1;
  take y;
  thus y in C & [x,y] in R & [y,z] in R by A6,A7,A8;
  x <= y by A7,WAYBEL_4:def 3;
  hence thesis by A9,ORDERS_2:def 6;
end;
