
theorem Th15:

:: 1.3. LEMMA, p. 143 (variant I)
  for S,T being lower-bounded non empty Poset for S9 being lower
  correct TopAugmentation of S for T9 being lower correct TopAugmentation of T
  holds omega [:S,T:] = the topology of [:S9,T9 qua non empty TopSpace:]
proof
  let S,T be lower-bounded non empty Poset;
  set ST = the lower correct TopAugmentation of [:S,T:];
  reconsider BX = the set of all (uparrow x)` where x is Element of ST
  as prebasis of ST by Def1;
  let S9 be lower correct TopAugmentation of S;
  reconsider BS = the set of all (uparrow x)` where x is Element of S9
  as prebasis of S9 by Def1;
  let T9 be lower correct TopAugmentation of T;
  set SxT = [:S9,T9 qua non empty TopSpace:];
  set B2 = {[:a, the carrier of T9:] where a is Subset of S9: a in BS};
A1: the RelStr of T9 = the RelStr of T by YELLOW_9:def 4;
  reconsider BT = the set of all (uparrow x)` where x is Element of T9
  as prebasis of T9 by Def1;
A2: the RelStr of S9 = the RelStr of S by YELLOW_9:def 4;
  then
A3: the carrier of SxT = [:the carrier of S, the carrier of T:] by A1,
BORSUK_1:def 2;
A4: the RelStr of ST = [:S,T:] by YELLOW_9:def 4;
  then
A5: the carrier of ST = [:the carrier of S, the carrier of T:] by
YELLOW_3:def 2;
A6: BX c= the topology of SxT
  proof
    let z be object;
A7: [#] T9 is open;
    assume z in BX;
    then consider x being Element of ST such that
A8: z = (uparrow x)`;
    consider s,t being object such that
A9: s in the carrier of S and
A10: t in the carrier of T and
A11: x = [s,t] by A5,ZFMISC_1:def 2;
    reconsider t as Element of T by A10;
    reconsider s as Element of S by A9;
    uparrow x = uparrow [s,t] by A4,A11,WAYBEL_0:13;
    then
A12: z = [:(uparrow s)`, [#]T:] \/ [:[#]S, (uparrow t)`:] by A4,A8,Th14;
    reconsider s9 = s as Element of S9 by A2;
    reconsider t9 = t as Element of T9 by A1;
    (uparrow t9)` in BT;
    then
A13: (uparrow t9)` is open by TOPS_2:def 1;
    reconsider A1 = [:(uparrow s)`, [#]T:], A2 = [:[#]S, (uparrow t)`:] as
    Subset of SxT by A3,YELLOW_3:def 2;
A14: [#]S9 is open;
    (uparrow s9)` in BS;
    then
A15: (uparrow s9)` is open by TOPS_2:def 1;
    uparrow t = uparrow t9 by A1,WAYBEL_0:13;
    then
A16: A2 is open by A13,A14,A2,A1,BORSUK_1:6;
    uparrow s = uparrow s9 by A2,WAYBEL_0:13;
    then A1 is open by A15,A7,A2,A1,BORSUK_1:6;
    then A1 \/ A2 is open by A16;
    hence thesis by A12;
  end;
  set B1 = {[:the carrier of S9, b:] where b is Subset of T9: b in BT};
  reconsider BB = B1 \/ B2 as prebasis of SxT by YELLOW_9:41;
A17: UniCl the topology of SxT = the topology of SxT by CANTOR_1:6;
  BB c= BX
  proof
    let z be object;
    assume
A18: z in BB;
    per cases by A18,XBOOLE_0:def 3;
    suppose
      z in B1;
      then consider b being Subset of T9 such that
A19:  z = [:the carrier of S9, b:] and
A20:  b in BT;
      consider t9 being Element of T9 such that
A21:  b = (uparrow t9)` by A20;
      reconsider t = t9 as Element of T by A1;
      reconsider x = [Bottom S,t] as Element of ST by A4;
A22:  uparrow x = uparrow [Bottom S,t] by A4,WAYBEL_0:13;
      uparrow Bottom S = the carrier of S by WAYBEL14:10;
      then
A23:  (uparrow Bottom S)` = {} by XBOOLE_1:37;
      uparrow t = uparrow t9 by A1,WAYBEL_0:13;
      then
      (uparrow [Bottom S,t])` = [:{}, the carrier of T:] \/ [:the carrier
      of S, b:] by A23,A1,A21,Th14
        .= {} \/ [:the carrier of S, b:] by ZFMISC_1:90
        .= z by A2,A19;
      hence thesis by A4,A22;
    end;
    suppose
      z in B2;
      then consider a being Subset of S9 such that
A24:  z = [:a, the carrier of T9:] and
A25:  a in BS;
      consider s9 being Element of S9 such that
A26:  a = (uparrow s9)` by A25;
      reconsider s = s9 as Element of S by A2;
      reconsider x = [s,Bottom T] as Element of ST by A4;
A27:  uparrow x = uparrow [s,Bottom T] by A4,WAYBEL_0:13;
      uparrow Bottom T = the carrier of T by WAYBEL14:10;
      then
A28:  (uparrow Bottom T)` = {} by XBOOLE_1:37;
      uparrow s = uparrow s9 by A2,WAYBEL_0:13;
      then (uparrow [s,Bottom T])` = [:a, the carrier of T:] \/ [:the carrier
      of S, {}:] by A28,A2,A26,Th14
        .= [:a, the carrier of T:] \/ {} by ZFMISC_1:90
        .= z by A1,A24;
      hence thesis by A4,A27;
    end;
  end;
  then FinMeetCl BB c= FinMeetCl BX by A5,A3,CANTOR_1:14;
  then
A29: UniCl FinMeetCl BB c= UniCl FinMeetCl BX by A5,A3,CANTOR_1:9;
  FinMeetCl BB is Basis of SxT by YELLOW_9:23;
  then
A30: the topology of SxT = UniCl FinMeetCl BB by YELLOW_9:22;
  FinMeetCl BX is Basis of ST by YELLOW_9:23;
  then
A31: the topology of ST = UniCl FinMeetCl BX by YELLOW_9:22;
  FinMeetCl the topology of SxT = the topology of SxT by CANTOR_1:5;
  then FinMeetCl BX c= the topology of SxT by A6,A5,A3,CANTOR_1:14;
  then UniCl FinMeetCl BX c= the topology of SxT by A5,A3,A17,CANTOR_1:9;
  then the topology of ST = the topology of SxT by A31,A30,A29;
  hence thesis by Def2;
end;
