reserve x,y for Real,
  u,v,w for set,
  r for positive Real;

theorem Th74:
  {[.0,a.[ where a is Real: 0 < a & a <= 1} \/ {].a,1.] where a is
  Real: 0 <= a & a < 1} \/
   {].a,b.[ where a,b is Real: 0 <= a & a < b & b <= 1}
  is Basis of I[01]
proof
  reconsider U = [.0,2/3.[, V = ].1/3,1.] as Subset of I[01] by BORSUK_1:40
,XXREAL_1:35,36;
  reconsider B = {].a,b.[ where a,b is Real:
   a < b} as Basis of R^1 by Th72;
  set S = I[01], T = R^1;
  set A1 = {[.0,a.[ where a is Real: 0 < a & a <= 1};
  set A2 = {].a,1.] where a is Real: 0 <= a & a < 1};
  set A3 = {].a,b.[ where a,b is Real: 0 <= a & a < b & b <= 1};
  set B9 = {A/\[#]S where A is Subset of T: A in B & A meets [#]S};
A1: B9 = A1 \/ A2 \/ A3 \/ {[#]S}
  proof
    reconsider aa = ].-1,2.[ as Subset of T by TOPMETR:17;
    thus B9 c= A1 \/ A2 \/ A3 \/ {[#]S}
    proof
      let u be object;
      assume u in B9;
      then consider A being Subset of T such that
A2:   u = A/\[#]S and
A3:   A in B and
A4:   A meets [#]S;
      consider x being object such that
A5:   x in A and
A6:   x in [#]S by A4,XBOOLE_0:3;
      consider a,b being Real such that
A7:   A = ].a,b.[ and
A8:   a < b by A3;
      reconsider x as Real by A5;
A9:   a < x by A7,A5,XXREAL_1:4;
A10:  x <= 1 by A6,BORSUK_1:40,XXREAL_1:1;
A11:  0 <= x by A6,BORSUK_1:40,XXREAL_1:1;
      per cases;
      suppose
A12:    a < 0 & b <= 1;
A13:    0 < b by A11,A7,A5,XXREAL_1:4;
        u = [.0,b.[ by A12,A2,A7,BORSUK_1:40,XXREAL_1:148;
        then u in A1 by A13,A12;
        then u in A1 \/ A2 by XBOOLE_0:def 3;
        then u in A1 \/ A2 \/ (A3 \/ {[#]S}) by XBOOLE_0:def 3;
        hence thesis by XBOOLE_1:4;
      end;
      suppose
        a < 0 & b > 1;
        then u = [#]S by A2,A7,BORSUK_1:40,XBOOLE_1:28,XXREAL_1:47;
        hence thesis by ZFMISC_1:136;
      end;
      suppose
A14:    a >= 0 & b <= 1;
        then u = A by A2,A7,BORSUK_1:40,XBOOLE_1:28,XXREAL_1:37;
        then u in A3 by A7,A8,A14;
        then u in A1 \/ A2 \/ A3 by XBOOLE_0:def 3;
        hence thesis by XBOOLE_0:def 3;
      end;
      suppose
A15:    a >= 0 & b > 1;
A16:    a < 1 by A10,A9,XXREAL_0:2;
        u = ].a,1.] by A15,A2,A7,BORSUK_1:40,XXREAL_1:149;
        then u in A2 by A16,A15;
        then u in A1 \/ A2 by XBOOLE_0:def 3;
        then u in A1 \/ A2 \/ (A3 \/ {[#]S}) by XBOOLE_0:def 3;
        hence thesis by XBOOLE_1:4;
      end;
    end;
    let u be object;
    assume u in A1 \/ A2 \/ A3 \/ {[#]S};
    then u in A1 \/ A2 \/ A3 or u in {[#]S} by XBOOLE_0:def 3;
    then
A17: u in A1 \/ A2 or u in A3 or u in {[#]S} by XBOOLE_0:def 3;
    per cases by A17,XBOOLE_0:def 3;
    suppose
      u in {[#]S};
      then u = [#]S by TARSKI:def 1;
      then
A18:  u = aa/\[#]S by BORSUK_1:40,XBOOLE_1:28,XXREAL_1:47;
      [#]S c= aa by BORSUK_1:40,XXREAL_1:47;
      then
A19:  aa meets [#]S by XBOOLE_1:68;
      aa in B;
      hence thesis by A18,A19;
    end;
    suppose
      u in A1;
      then consider a being Real such that
A20:  u = [.0,a.[ and
A21:  0 < a and
A22:  a <= 1;
      reconsider A = ].-1,a.[ as Subset of T by TOPMETR:17;
A23:  A in B by A21;
A24:  0 in [.0,1.] by XXREAL_1:1;
      0 in A by A21,XXREAL_1:4;
      then
A25:  A meets [#]S by A24,BORSUK_1:40,XBOOLE_0:3;
      u = A /\ [.0,1.] by A20,A22,XXREAL_1:148;
      hence thesis by A23,A25,BORSUK_1:40;
    end;
    suppose
      u in A2;
      then consider a being Real such that
A26:  u = ].a,1.] and
A27:  0 <= a and
A28:  a < 1;
      reconsider A = ].a,2.[ as Subset of T by TOPMETR:17;
      2 > a by A28,XXREAL_0:2;
      then
A29:  A in B;
A30:  1 in [.0,1.] by XXREAL_1:1;
      1 in A by A28,XXREAL_1:4;
      then
A31:  A meets [#]S by A30,BORSUK_1:40,XBOOLE_0:3;
      u = A /\ [.0,1.] by A26,A27,XXREAL_1:149;
      hence thesis by A29,A31,BORSUK_1:40;
    end;
    suppose
      u in A3;
      then consider a,b being Real such that
A32:  u = ].a,b.[ and
A33:  0 <= a and
A34:  a < b and
A35:  b <= 1;
      reconsider A = ].a,b.[ as Subset of T by TOPMETR:17;
A36:  A c= [.0,1.] by A33,A35,XXREAL_1:37;
      a+b < b+b by A34,XREAL_1:8;
      then
A37:  (a+b)/2 < (b+b)/2 by XREAL_1:74;
      a+a < a+b by A34,XREAL_1:8;
      then (a+a)/2 < (a+b)/2 by XREAL_1:74;
      then (a+b)/2 in A by A37,XXREAL_1:4;
      then
A38:  A meets [#]S by A36,BORSUK_1:40,XBOOLE_0:3;
A39:  A in B by A34;
      u = A /\ [.0,1.] by A33,A35,A32,XBOOLE_1:28,XXREAL_1:37;
      hence thesis by A39,A38,BORSUK_1:40;
    end;
  end;
  U in A1;
  then U in A1 \/ A2 by XBOOLE_0:def 3;
  then
A40: U in A1 \/ A2 \/ A3 by XBOOLE_0:def 3;
  V in A2;
  then V in A1 \/ A2 by XBOOLE_0:def 3;
  then
A41: V in A1 \/ A2 \/ A3 by XBOOLE_0:def 3;
  U \/ V = [#]S by BORSUK_1:40,XXREAL_1:175;
  hence thesis by A1,A40,A41,Th71,Th73;
end;
