
theorem
  for L being lower-bounded with_suprema Poset, R being auxiliary(i)
auxiliary(ii) (Relation of L), C being strict_chain of R st C is maximal & R is
  satisfying_SI holds R satisfies_SIC_on C
proof
  let L be lower-bounded with_suprema Poset, R be auxiliary(i) auxiliary(ii) (
  Relation of L), C be strict_chain of R such that
A1: C is maximal and
A2: R is satisfying_SI;
  let x, z be Element of L;
  assume that
A3: x in C and
A4: z in C and
A5: [x,z] in R and
A6: x <> z;
  consider y being Element of L such that
A7: [x,y] in R and
A8: [y,z] in R and
A9: x <> y by A2,A5,A6,WAYBEL_4:def 20;
A10: y <= z by A8,WAYBEL_4:def 3;
  assume
A11: not thesis;
A12: x <= y by A7,WAYBEL_4:def 3;
A13: C \/ {y} is strict_chain of R
  proof
    let a, b be set such that
A14: a in C \/ {y} and
A15: b in C \/ {y};
    per cases by A14,A15,Lm1;
    suppose
      a in C & b in C;
      hence thesis by Def3;
    end;
    suppose that
A16:  a in C and
A17:  b = y;
      now
        reconsider a as Element of L by A16;
A18:    a <= a;
        per cases by A11,A16;
        suppose
          x = a;
          hence thesis by A7,A17;
        end;
        suppose
          a = z;
          hence thesis by A8,A17;
        end;
        suppose
          not [x,a] in R & a <> x;
          then [a,x] in R by A3,A16,Def3;
          hence thesis by A12,A17,A18,WAYBEL_4:def 4;
        end;
        suppose
          not [a,z] in R & a <> z;
          then [z,a] in R by A4,A16,Def3;
          hence thesis by A10,A17,A18,WAYBEL_4:def 4;
        end;
      end;
      hence thesis;
    end;
    suppose that
A19:  a = y and
A20:  b in C;
      now
        reconsider b as Element of L by A20;
A21:    b <= b;
        per cases by A11,A20;
        suppose
          x = b;
          hence thesis by A7,A19;
        end;
        suppose
          b = z;
          hence thesis by A8,A19;
        end;
        suppose
          not [x,b] in R & b <> x;
          then [b,x] in R by A3,A20,Def3;
          hence thesis by A12,A19,A21,WAYBEL_4:def 4;
        end;
        suppose
          not [b,z] in R & b <> z;
          then [z,b] in R by A4,A20,Def3;
          hence thesis by A10,A19,A21,WAYBEL_4:def 4;
        end;
      end;
      hence thesis;
    end;
    suppose
      a = y & b = y;
      hence thesis;
    end;
  end;
  C c= C \/ {y} by XBOOLE_1:7;
  then C \/ {y} = C by A13,A1;
  then y in C by ZFMISC_1:39;
  hence thesis by A11,A7,A8,A9;
end;
