set B = {0,1} \/ ].(1 / 2),1.[;

set L = RealSubLatt ((In (0,REAL)),(In (1,REAL)));

set R = Real_Lattice ;

A1: {0,1} \/ ].(1 / 2),1.[ c= {0} \/ { x where x is Real : ( 1 / 2 < x & x <= 1 ) }

A10: 0 in { r where r is Real : ( 0 <= r & r <= 1 ) } ;

then reconsider A = [.0,jj.] as non empty set by RCOMP_1:def 1;

A11: the L_meet of (RealSubLatt ((In (0,REAL)),(In (1,REAL)))) = minreal || [.0,jj.] by Def4;

reconsider B = {0,1} \/ ].(1 / 2),1.[ as non empty set ;

A12: the L_join of (RealSubLatt ((In (0,REAL)),(In (1,REAL)))) = maxreal || [.0,jj.] by Def4;

A13: A = the carrier of (RealSubLatt ((In (0,REAL)),(In (1,REAL)))) by Def4;

then reconsider Ma = maxreal || A, Mi = minreal || A as BinOp of A by Def4;

then A17: [:B,B:] c= [:A,A:] by ZFMISC_1:96;

then reconsider ma = Ma || B, mi = Mi || B as Function of [:B,B:],A by FUNCT_2:32;

A18: for x1 being Element of A holds x1 is Element of Real_Lattice by REAL_LAT:def 3, XREAL_0:def 1;

A19: dom mi = [:B,B:] by FUNCT_2:def 1;

reconsider ma = ma as BinOp of B by A34, A35, FUNCT_2:3;

reconsider L9 = LattStr(# B,ma,mi #) as non empty LattStr ;

then reconsider L9 = L9 as Lattice ;

reconsider L9 = L9 as SubLattice of RealSubLatt ((In (0,REAL)),(In (1,REAL))) by A13, A12, A11, A16, NAT_LAT:def 12;

take L9 ; :: thesis: ( L9 is \/-inheriting & not L9 is /\-inheriting )

set L = RealSubLatt ((In (0,REAL)),(In (1,REAL)));

set R = Real_Lattice ;

A1: {0,1} \/ ].(1 / 2),1.[ c= {0} \/ { x where x is Real : ( 1 / 2 < x & x <= 1 ) }

proof

{0} \/ { x where x is Real : ( 1 / 2 < x & x <= 1 ) } c= {0,1} \/ ].(1 / 2),1.[
let x1 be object ; :: according to TARSKI:def 3 :: thesis: ( not x1 in {0,1} \/ ].(1 / 2),1.[ or x1 in {0} \/ { x where x is Real : ( 1 / 2 < x & x <= 1 ) } )

assume A2: x1 in {0,1} \/ ].(1 / 2),1.[ ; :: thesis: x1 in {0} \/ { x where x is Real : ( 1 / 2 < x & x <= 1 ) }

end;assume A2: x1 in {0,1} \/ ].(1 / 2),1.[ ; :: thesis: x1 in {0} \/ { x where x is Real : ( 1 / 2 < x & x <= 1 ) }

now :: thesis: x1 in {0} \/ { x where x is Real : ( 1 / 2 < x & x <= 1 ) } end;

hence
x1 in {0} \/ { x where x is Real : ( 1 / 2 < x & x <= 1 ) }
; :: thesis: verumper cases
( x1 in {0,1} or x1 in ].(1 / 2),1.[ )
by A2, XBOOLE_0:def 3;

end;

suppose
x1 in {0,1}
; :: thesis: x1 in {0} \/ { x where x is Real : ( 1 / 2 < x & x <= 1 ) }

then
( x1 = 0 or x1 = 1 )
by TARSKI:def 2;

then ( x1 in {0} or x1 in { x where x is Real : ( 1 / 2 < x & x <= jj ) } ) by TARSKI:def 1;

hence x1 in {0} \/ { x where x is Real : ( 1 / 2 < x & x <= 1 ) } by XBOOLE_0:def 3; :: thesis: verum

end;then ( x1 in {0} or x1 in { x where x is Real : ( 1 / 2 < x & x <= jj ) } ) by TARSKI:def 1;

hence x1 in {0} \/ { x where x is Real : ( 1 / 2 < x & x <= 1 ) } by XBOOLE_0:def 3; :: thesis: verum

suppose
x1 in ].(1 / 2),1.[
; :: thesis: x1 in {0} \/ { x where x is Real : ( 1 / 2 < x & x <= 1 ) }

then
x1 in { r where r is Real : ( 1 / 2 < r & r < 1 ) }
by RCOMP_1:def 2;

then consider y being Real such that

A3: x1 = y and

A4: ( 1 / 2 < y & y < 1 ) ;

y in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } by A4;

hence x1 in {0} \/ { x where x is Real : ( 1 / 2 < x & x <= 1 ) } by A3, XBOOLE_0:def 3; :: thesis: verum

end;then consider y being Real such that

A3: x1 = y and

A4: ( 1 / 2 < y & y < 1 ) ;

y in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } by A4;

hence x1 in {0} \/ { x where x is Real : ( 1 / 2 < x & x <= 1 ) } by A3, XBOOLE_0:def 3; :: thesis: verum

proof

then A9:
{0,1} \/ ].(1 / 2),1.[ = {0} \/ { x where x is Real : ( 1 / 2 < x & x <= 1 ) }
by A1, XBOOLE_0:def 10;
let x1 be object ; :: according to TARSKI:def 3 :: thesis: ( not x1 in {0} \/ { x where x is Real : ( 1 / 2 < x & x <= 1 ) } or x1 in {0,1} \/ ].(1 / 2),1.[ )

assume A5: x1 in {0} \/ { x where x is Real : ( 1 / 2 < x & x <= 1 ) } ; :: thesis: x1 in {0,1} \/ ].(1 / 2),1.[

end;assume A5: x1 in {0} \/ { x where x is Real : ( 1 / 2 < x & x <= 1 ) } ; :: thesis: x1 in {0,1} \/ ].(1 / 2),1.[

now :: thesis: x1 in {0,1} \/ ].(1 / 2),1.[end;

hence
x1 in {0,1} \/ ].(1 / 2),1.[
; :: thesis: verumper cases
( x1 in {0} or x1 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } )
by A5, XBOOLE_0:def 3;

end;

suppose
x1 in {0}
; :: thesis: x1 in {0,1} \/ ].(1 / 2),1.[

then
x1 = 0
by TARSKI:def 1;

then x1 in {0,1} by TARSKI:def 2;

hence x1 in {0,1} \/ ].(1 / 2),1.[ by XBOOLE_0:def 3; :: thesis: verum

end;then x1 in {0,1} by TARSKI:def 2;

hence x1 in {0,1} \/ ].(1 / 2),1.[ by XBOOLE_0:def 3; :: thesis: verum

suppose
x1 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) }
; :: thesis: x1 in {0,1} \/ ].(1 / 2),1.[

then consider y being Real such that

A6: x1 = y and

A7: 1 / 2 < y and

A8: y <= 1 ;

( y < 1 or y = 1 ) by A8, XXREAL_0:1;

then ( y in { r where r is Real : ( 1 / 2 < r & r < 1 ) } or y = 1 ) by A7;

then ( y in ].(1 / 2),1.[ or y in {0,1} ) by RCOMP_1:def 2, TARSKI:def 2;

hence x1 in {0,1} \/ ].(1 / 2),1.[ by A6, XBOOLE_0:def 3; :: thesis: verum

end;A6: x1 = y and

A7: 1 / 2 < y and

A8: y <= 1 ;

( y < 1 or y = 1 ) by A8, XXREAL_0:1;

then ( y in { r where r is Real : ( 1 / 2 < r & r < 1 ) } or y = 1 ) by A7;

then ( y in ].(1 / 2),1.[ or y in {0,1} ) by RCOMP_1:def 2, TARSKI:def 2;

hence x1 in {0,1} \/ ].(1 / 2),1.[ by A6, XBOOLE_0:def 3; :: thesis: verum

A10: 0 in { r where r is Real : ( 0 <= r & r <= 1 ) } ;

then reconsider A = [.0,jj.] as non empty set by RCOMP_1:def 1;

A11: the L_meet of (RealSubLatt ((In (0,REAL)),(In (1,REAL)))) = minreal || [.0,jj.] by Def4;

reconsider B = {0,1} \/ ].(1 / 2),1.[ as non empty set ;

A12: the L_join of (RealSubLatt ((In (0,REAL)),(In (1,REAL)))) = maxreal || [.0,jj.] by Def4;

A13: A = the carrier of (RealSubLatt ((In (0,REAL)),(In (1,REAL)))) by Def4;

then reconsider Ma = maxreal || A, Mi = minreal || A as BinOp of A by Def4;

A14: now :: thesis: for x1 being object st x1 in B holds

x1 in A

then A16:
B c= A
;x1 in A

let x1 be object ; :: thesis: ( x1 in B implies x1 in A )

assume A15: x1 in B ; :: thesis: x1 in A

end;assume A15: x1 in B ; :: thesis: x1 in A

now :: thesis: x1 in A

hence
x1 in A
; :: thesis: verumend;

then A17: [:B,B:] c= [:A,A:] by ZFMISC_1:96;

then reconsider ma = Ma || B, mi = Mi || B as Function of [:B,B:],A by FUNCT_2:32;

A18: for x1 being Element of A holds x1 is Element of Real_Lattice by REAL_LAT:def 3, XREAL_0:def 1;

A19: dom mi = [:B,B:] by FUNCT_2:def 1;

A20: now :: thesis: for x9 being object st x9 in dom mi holds

mi . x9 in B

A34:
dom ma = [:B,B:]
by FUNCT_2:def 1;mi . x9 in B

let x9 be object ; :: thesis: ( x9 in dom mi implies mi . x9 in B )

assume A21: x9 in dom mi ; :: thesis: mi . x9 in B

then consider x1, x2 being object such that

A22: x9 = [x1,x2] by RELAT_1:def 1;

A23: x2 in B by A21, A22, ZFMISC_1:87;

A24: x1 in B by A21, A22, ZFMISC_1:87;

end;assume A21: x9 in dom mi ; :: thesis: mi . x9 in B

then consider x1, x2 being object such that

A22: x9 = [x1,x2] by RELAT_1:def 1;

A23: x2 in B by A21, A22, ZFMISC_1:87;

A24: x1 in B by A21, A22, ZFMISC_1:87;

now :: thesis: mi . x9 in Bend;

hence
mi . x9 in B
; :: thesis: verumper cases
( x1 in {0} or x1 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } )
by A1, A24, XBOOLE_0:def 3;

end;

suppose
x1 in {0}
; :: thesis: mi . x9 in B

then A25:
x1 = 0
by TARSKI:def 1;

end;now :: thesis: mi . x9 in Bend;

hence
mi . x9 in B
; :: thesis: verumper cases
( x2 in {0} or x2 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } )
by A1, A23, XBOOLE_0:def 3;

end;

suppose
x2 in {0}
; :: thesis: mi . x9 in B

then A26:
x2 = 0
by TARSKI:def 1;

mi . x9 = Mi . [x1,x2] by A21, A22, FUNCT_1:49

.= minreal . (x1,x2) by A17, A21, A22, FUNCT_1:49

.= min (0,0) by A25, A26, REAL_LAT:def 1

.= 0 ;

then mi . x9 in {0} by TARSKI:def 1;

hence mi . x9 in B by A9, XBOOLE_0:def 3; :: thesis: verum

end;mi . x9 = Mi . [x1,x2] by A21, A22, FUNCT_1:49

.= minreal . (x1,x2) by A17, A21, A22, FUNCT_1:49

.= min (0,0) by A25, A26, REAL_LAT:def 1

.= 0 ;

then mi . x9 in {0} by TARSKI:def 1;

hence mi . x9 in B by A9, XBOOLE_0:def 3; :: thesis: verum

suppose
x2 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) }
; :: thesis: mi . x9 in B

then consider y2 being Real such that

A27: x2 = y2 and

A28: ( 1 / 2 < y2 & y2 <= 1 ) ;

reconsider y2 = y2 as Real ;

mi . x9 = Mi . [x1,x2] by A21, A22, FUNCT_1:49

.= minreal . (x1,x2) by A17, A21, A22, FUNCT_1:49

.= min (0,y2) by A25, A27, REAL_LAT:def 1 ;

then ( mi . x9 = 0 or mi . x9 = y2 ) by XXREAL_0:15;

then ( mi . x9 in {0} or mi . x9 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } ) by A28, TARSKI:def 1;

hence mi . x9 in B by A9, XBOOLE_0:def 3; :: thesis: verum

end;A27: x2 = y2 and

A28: ( 1 / 2 < y2 & y2 <= 1 ) ;

reconsider y2 = y2 as Real ;

mi . x9 = Mi . [x1,x2] by A21, A22, FUNCT_1:49

.= minreal . (x1,x2) by A17, A21, A22, FUNCT_1:49

.= min (0,y2) by A25, A27, REAL_LAT:def 1 ;

then ( mi . x9 = 0 or mi . x9 = y2 ) by XXREAL_0:15;

then ( mi . x9 in {0} or mi . x9 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } ) by A28, TARSKI:def 1;

hence mi . x9 in B by A9, XBOOLE_0:def 3; :: thesis: verum

suppose
x1 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) }
; :: thesis: mi . x9 in B

then consider y1 being Real such that

A29: x1 = y1 and

A30: ( 1 / 2 < y1 & y1 <= 1 ) ;

reconsider y1 = y1 as Real ;

end;A29: x1 = y1 and

A30: ( 1 / 2 < y1 & y1 <= 1 ) ;

reconsider y1 = y1 as Real ;

now :: thesis: mi . x9 in Bend;

hence
mi . x9 in B
; :: thesis: verumper cases
( x2 in {0} or x2 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } )
by A1, A23, XBOOLE_0:def 3;

end;

suppose
x2 in {0}
; :: thesis: mi . x9 in B

then A31:
x2 = 0
by TARSKI:def 1;

mi . x9 = Mi . [x1,x2] by A21, A22, FUNCT_1:49

.= minreal . (x1,x2) by A17, A21, A22, FUNCT_1:49

.= min (y1,0) by A29, A31, REAL_LAT:def 1 ;

then ( mi . x9 = y1 or mi . x9 = 0 ) by XXREAL_0:15;

then ( mi . x9 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } or mi . x9 in {0} ) by A30, TARSKI:def 1;

hence mi . x9 in B by A9, XBOOLE_0:def 3; :: thesis: verum

end;mi . x9 = Mi . [x1,x2] by A21, A22, FUNCT_1:49

.= minreal . (x1,x2) by A17, A21, A22, FUNCT_1:49

.= min (y1,0) by A29, A31, REAL_LAT:def 1 ;

then ( mi . x9 = y1 or mi . x9 = 0 ) by XXREAL_0:15;

then ( mi . x9 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } or mi . x9 in {0} ) by A30, TARSKI:def 1;

hence mi . x9 in B by A9, XBOOLE_0:def 3; :: thesis: verum

suppose
x2 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) }
; :: thesis: mi . x9 in B

then consider y2 being Real such that

A32: x2 = y2 and

A33: ( 1 / 2 < y2 & y2 <= 1 ) ;

reconsider y2 = y2 as Real ;

mi . x9 = Mi . [x1,x2] by A21, A22, FUNCT_1:49

.= minreal . (x1,x2) by A17, A21, A22, FUNCT_1:49

.= min (y1,y2) by A29, A32, REAL_LAT:def 1 ;

then ( mi . x9 = y1 or mi . x9 = y2 ) by XXREAL_0:15;

then mi . x9 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } by A30, A33;

hence mi . x9 in B by A9, XBOOLE_0:def 3; :: thesis: verum

end;A32: x2 = y2 and

A33: ( 1 / 2 < y2 & y2 <= 1 ) ;

reconsider y2 = y2 as Real ;

mi . x9 = Mi . [x1,x2] by A21, A22, FUNCT_1:49

.= minreal . (x1,x2) by A17, A21, A22, FUNCT_1:49

.= min (y1,y2) by A29, A32, REAL_LAT:def 1 ;

then ( mi . x9 = y1 or mi . x9 = y2 ) by XXREAL_0:15;

then mi . x9 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } by A30, A33;

hence mi . x9 in B by A9, XBOOLE_0:def 3; :: thesis: verum

A35: now :: thesis: for x9 being object st x9 in dom ma holds

ma . x9 in B

reconsider mi = mi as BinOp of B by A19, A20, FUNCT_2:3;ma . x9 in B

let x9 be object ; :: thesis: ( x9 in dom ma implies ma . x9 in B )

assume A36: x9 in dom ma ; :: thesis: ma . x9 in B

then consider x1, x2 being object such that

A37: x9 = [x1,x2] by RELAT_1:def 1;

A38: x2 in B by A36, A37, ZFMISC_1:87;

A39: x1 in B by A36, A37, ZFMISC_1:87;

end;assume A36: x9 in dom ma ; :: thesis: ma . x9 in B

then consider x1, x2 being object such that

A37: x9 = [x1,x2] by RELAT_1:def 1;

A38: x2 in B by A36, A37, ZFMISC_1:87;

A39: x1 in B by A36, A37, ZFMISC_1:87;

now :: thesis: ma . x9 in Bend;

hence
ma . x9 in B
; :: thesis: verumper cases
( x1 in {0} or x1 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } )
by A1, A39, XBOOLE_0:def 3;

end;

suppose
x1 in {0}
; :: thesis: ma . x9 in B

then A40:
x1 = 0
by TARSKI:def 1;

end;now :: thesis: ma . x9 in Bend;

hence
ma . x9 in B
; :: thesis: verumper cases
( x2 in {0} or x2 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } )
by A1, A38, XBOOLE_0:def 3;

end;

suppose
x2 in {0}
; :: thesis: ma . x9 in B

then A41:
x2 = 0
by TARSKI:def 1;

ma . x9 = Ma . [x1,x2] by A36, A37, FUNCT_1:49

.= maxreal . (x1,x2) by A17, A36, A37, FUNCT_1:49

.= max (0,0) by A40, A41, REAL_LAT:def 2

.= 0 ;

then ma . x9 in {0} by TARSKI:def 1;

hence ma . x9 in B by A9, XBOOLE_0:def 3; :: thesis: verum

end;ma . x9 = Ma . [x1,x2] by A36, A37, FUNCT_1:49

.= maxreal . (x1,x2) by A17, A36, A37, FUNCT_1:49

.= max (0,0) by A40, A41, REAL_LAT:def 2

.= 0 ;

then ma . x9 in {0} by TARSKI:def 1;

hence ma . x9 in B by A9, XBOOLE_0:def 3; :: thesis: verum

suppose
x2 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) }
; :: thesis: ma . x9 in B

then consider y2 being Real such that

A42: x2 = y2 and

A43: ( 1 / 2 < y2 & y2 <= 1 ) ;

reconsider y2 = y2 as Real ;

ma . x9 = Ma . [x1,x2] by A36, A37, FUNCT_1:49

.= maxreal . (x1,x2) by A17, A36, A37, FUNCT_1:49

.= max (0,y2) by A40, A42, REAL_LAT:def 2 ;

then ( ma . x9 = 0 or ma . x9 = y2 ) by XXREAL_0:16;

then ( ma . x9 in {0} or ma . x9 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } ) by A43, TARSKI:def 1;

hence ma . x9 in B by A9, XBOOLE_0:def 3; :: thesis: verum

end;A42: x2 = y2 and

A43: ( 1 / 2 < y2 & y2 <= 1 ) ;

reconsider y2 = y2 as Real ;

ma . x9 = Ma . [x1,x2] by A36, A37, FUNCT_1:49

.= maxreal . (x1,x2) by A17, A36, A37, FUNCT_1:49

.= max (0,y2) by A40, A42, REAL_LAT:def 2 ;

then ( ma . x9 = 0 or ma . x9 = y2 ) by XXREAL_0:16;

then ( ma . x9 in {0} or ma . x9 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } ) by A43, TARSKI:def 1;

hence ma . x9 in B by A9, XBOOLE_0:def 3; :: thesis: verum

suppose
x1 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) }
; :: thesis: ma . x9 in B

then consider y1 being Real such that

A44: x1 = y1 and

A45: ( 1 / 2 < y1 & y1 <= 1 ) ;

reconsider y1 = y1 as Real ;

end;A44: x1 = y1 and

A45: ( 1 / 2 < y1 & y1 <= 1 ) ;

reconsider y1 = y1 as Real ;

now :: thesis: ma . x9 in Bend;

hence
ma . x9 in B
; :: thesis: verumper cases
( x2 in {0} or x2 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } )
by A1, A38, XBOOLE_0:def 3;

end;

suppose
x2 in {0}
; :: thesis: ma . x9 in B

then A46:
x2 = 0
by TARSKI:def 1;

ma . x9 = Ma . [x1,x2] by A36, A37, FUNCT_1:49

.= maxreal . (x1,x2) by A17, A36, A37, FUNCT_1:49

.= max (y1,0) by A44, A46, REAL_LAT:def 2 ;

then ( ma . x9 = y1 or ma . x9 = 0 ) by XXREAL_0:16;

then ( ma . x9 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } or ma . x9 in {0} ) by A45, TARSKI:def 1;

hence ma . x9 in B by A9, XBOOLE_0:def 3; :: thesis: verum

end;ma . x9 = Ma . [x1,x2] by A36, A37, FUNCT_1:49

.= maxreal . (x1,x2) by A17, A36, A37, FUNCT_1:49

.= max (y1,0) by A44, A46, REAL_LAT:def 2 ;

then ( ma . x9 = y1 or ma . x9 = 0 ) by XXREAL_0:16;

then ( ma . x9 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } or ma . x9 in {0} ) by A45, TARSKI:def 1;

hence ma . x9 in B by A9, XBOOLE_0:def 3; :: thesis: verum

suppose
x2 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) }
; :: thesis: ma . x9 in B

then consider y2 being Real such that

A47: x2 = y2 and

A48: ( 1 / 2 < y2 & y2 <= 1 ) ;

reconsider y2 = y2 as Real ;

ma . x9 = Ma . [x1,x2] by A36, A37, FUNCT_1:49

.= maxreal . (x1,x2) by A17, A36, A37, FUNCT_1:49

.= max (y1,y2) by A44, A47, REAL_LAT:def 2 ;

then ( ma . x9 = y1 or ma . x9 = y2 ) by XXREAL_0:16;

then ma . x9 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } by A45, A48;

hence ma . x9 in B by A9, XBOOLE_0:def 3; :: thesis: verum

end;A47: x2 = y2 and

A48: ( 1 / 2 < y2 & y2 <= 1 ) ;

reconsider y2 = y2 as Real ;

ma . x9 = Ma . [x1,x2] by A36, A37, FUNCT_1:49

.= maxreal . (x1,x2) by A17, A36, A37, FUNCT_1:49

.= max (y1,y2) by A44, A47, REAL_LAT:def 2 ;

then ( ma . x9 = y1 or ma . x9 = y2 ) by XXREAL_0:16;

then ma . x9 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } by A45, A48;

hence ma . x9 in B by A9, XBOOLE_0:def 3; :: thesis: verum

reconsider ma = ma as BinOp of B by A34, A35, FUNCT_2:3;

reconsider L9 = LattStr(# B,ma,mi #) as non empty LattStr ;

A49: now :: thesis: for a, b being Element of L9 holds

( a "\/" b = maxreal . (a,b) & a "/\" b = minreal . (a,b) )

reconsider L = RealSubLatt ((In (0,REAL)),(In (1,REAL))) as complete Lattice by Th20;( a "\/" b = maxreal . (a,b) & a "/\" b = minreal . (a,b) )

let a, b be Element of L9; :: thesis: ( a "\/" b = maxreal . (a,b) & a "/\" b = minreal . (a,b) )

thus a "\/" b = the L_join of L9 . (a,b) by LATTICES:def 1

.= (maxreal || A) . [a,b] by FUNCT_1:49

.= maxreal . (a,b) by A17, FUNCT_1:49 ; :: thesis: a "/\" b = minreal . (a,b)

thus a "/\" b = the L_meet of L9 . (a,b) by LATTICES:def 2

.= (minreal || A) . [a,b] by FUNCT_1:49

.= minreal . (a,b) by A17, FUNCT_1:49 ; :: thesis: verum

end;thus a "\/" b = the L_join of L9 . (a,b) by LATTICES:def 1

.= (maxreal || A) . [a,b] by FUNCT_1:49

.= maxreal . (a,b) by A17, FUNCT_1:49 ; :: thesis: a "/\" b = minreal . (a,b)

thus a "/\" b = the L_meet of L9 . (a,b) by LATTICES:def 2

.= (minreal || A) . [a,b] by FUNCT_1:49

.= minreal . (a,b) by A17, FUNCT_1:49 ; :: thesis: verum

A50: now :: thesis: for x1 being Element of B holds x1 is Element of L

let x1 be Element of B; :: thesis: x1 is Element of L

end;now :: thesis: x1 is Element of L

hence
x1 is Element of L
; :: thesis: verumend;

A51: now :: thesis: for p, q being Element of L9 holds p "\/" q = q "\/" p

let p, q be Element of L9; :: thesis: p "\/" q = q "\/" p

reconsider p9 = p, q9 = q as Element of L by A50;

reconsider p9 = p9, q9 = q9 as Element of Real_Lattice by A13, A18;

thus p "\/" q = maxreal . (p,q) by A49

.= maxreal . (q9,p9) by REAL_LAT:1

.= q "\/" p by A49 ; :: thesis: verum

end;reconsider p9 = p, q9 = q as Element of L by A50;

reconsider p9 = p9, q9 = q9 as Element of Real_Lattice by A13, A18;

thus p "\/" q = maxreal . (p,q) by A49

.= maxreal . (q9,p9) by REAL_LAT:1

.= q "\/" p by A49 ; :: thesis: verum

A52: now :: thesis: for p, q being Element of L9 holds p "/\" (p "\/" q) = p

let p, q be Element of L9; :: thesis: p "/\" (p "\/" q) = p

reconsider p9 = p, q9 = q as Element of L by A50;

reconsider p9 = p9, q9 = q9 as Element of Real_Lattice by A13, A18;

thus p "/\" (p "\/" q) = minreal . (p,(p "\/" q)) by A49

.= minreal . (p9,(maxreal . (p9,q9))) by A49

.= p by REAL_LAT:6 ; :: thesis: verum

end;reconsider p9 = p, q9 = q as Element of L by A50;

reconsider p9 = p9, q9 = q9 as Element of Real_Lattice by A13, A18;

thus p "/\" (p "\/" q) = minreal . (p,(p "\/" q)) by A49

.= minreal . (p9,(maxreal . (p9,q9))) by A49

.= p by REAL_LAT:6 ; :: thesis: verum

A53: now :: thesis: for p, q being Element of L9 holds p "/\" q = q "/\" p

let p, q be Element of L9; :: thesis: p "/\" q = q "/\" p

reconsider p9 = p, q9 = q as Element of L by A50;

reconsider p9 = p9, q9 = q9 as Element of Real_Lattice by A13, A18;

thus p "/\" q = minreal . (p,q) by A49

.= minreal . (q9,p9) by REAL_LAT:2

.= q "/\" p by A49 ; :: thesis: verum

end;reconsider p9 = p, q9 = q as Element of L by A50;

reconsider p9 = p9, q9 = q9 as Element of Real_Lattice by A13, A18;

thus p "/\" q = minreal . (p,q) by A49

.= minreal . (q9,p9) by REAL_LAT:2

.= q "/\" p by A49 ; :: thesis: verum

A54: now :: thesis: for p, q being Element of L9 holds (p "/\" q) "\/" q = q

let p, q be Element of L9; :: thesis: (p "/\" q) "\/" q = q

reconsider p9 = p, q9 = q as Element of L by A50;

reconsider p9 = p9, q9 = q9 as Element of Real_Lattice by A13, A18;

thus (p "/\" q) "\/" q = maxreal . ((p "/\" q),q) by A49

.= maxreal . ((minreal . (p9,q9)),q9) by A49

.= q by REAL_LAT:5 ; :: thesis: verum

end;reconsider p9 = p, q9 = q as Element of L by A50;

reconsider p9 = p9, q9 = q9 as Element of Real_Lattice by A13, A18;

thus (p "/\" q) "\/" q = maxreal . ((p "/\" q),q) by A49

.= maxreal . ((minreal . (p9,q9)),q9) by A49

.= q by REAL_LAT:5 ; :: thesis: verum

A55: now :: thesis: for p, q, r being Element of L9 holds p "/\" (q "/\" r) = (p "/\" q) "/\" r

let p, q, r be Element of L9; :: thesis: p "/\" (q "/\" r) = (p "/\" q) "/\" r

reconsider p9 = p, q9 = q, r9 = r as Element of L by A50;

reconsider p9 = p9, q9 = q9, r9 = r9 as Element of Real_Lattice by A13, A18;

thus p "/\" (q "/\" r) = minreal . (p,(q "/\" r)) by A49

.= minreal . (p,(minreal . (q,r))) by A49

.= minreal . ((minreal . (p9,q9)),r9) by REAL_LAT:4

.= minreal . ((p "/\" q),r) by A49

.= (p "/\" q) "/\" r by A49 ; :: thesis: verum

end;reconsider p9 = p, q9 = q, r9 = r as Element of L by A50;

reconsider p9 = p9, q9 = q9, r9 = r9 as Element of Real_Lattice by A13, A18;

thus p "/\" (q "/\" r) = minreal . (p,(q "/\" r)) by A49

.= minreal . (p,(minreal . (q,r))) by A49

.= minreal . ((minreal . (p9,q9)),r9) by REAL_LAT:4

.= minreal . ((p "/\" q),r) by A49

.= (p "/\" q) "/\" r by A49 ; :: thesis: verum

now :: thesis: for p, q, r being Element of L9 holds p "\/" (q "\/" r) = (p "\/" q) "\/" r

then
( L9 is join-commutative & L9 is join-associative & L9 is meet-absorbing & L9 is meet-commutative & L9 is meet-associative & L9 is join-absorbing )
by A51, A54, A53, A55, A52, LATTICES:def 4, LATTICES:def 5, LATTICES:def 6, LATTICES:def 7, LATTICES:def 8, LATTICES:def 9;let p, q, r be Element of L9; :: thesis: p "\/" (q "\/" r) = (p "\/" q) "\/" r

reconsider p9 = p, q9 = q, r9 = r as Element of L by A50;

reconsider p9 = p9, q9 = q9, r9 = r9 as Element of Real_Lattice by A13, A18;

thus p "\/" (q "\/" r) = maxreal . (p,(q "\/" r)) by A49

.= maxreal . (p,(maxreal . (q,r))) by A49

.= maxreal . ((maxreal . (p9,q9)),r9) by REAL_LAT:3

.= maxreal . ((p "\/" q),r) by A49

.= (p "\/" q) "\/" r by A49 ; :: thesis: verum

end;reconsider p9 = p, q9 = q, r9 = r as Element of L by A50;

reconsider p9 = p9, q9 = q9, r9 = r9 as Element of Real_Lattice by A13, A18;

thus p "\/" (q "\/" r) = maxreal . (p,(q "\/" r)) by A49

.= maxreal . (p,(maxreal . (q,r))) by A49

.= maxreal . ((maxreal . (p9,q9)),r9) by REAL_LAT:3

.= maxreal . ((p "\/" q),r) by A49

.= (p "\/" q) "\/" r by A49 ; :: thesis: verum

then reconsider L9 = L9 as Lattice ;

reconsider L9 = L9 as SubLattice of RealSubLatt ((In (0,REAL)),(In (1,REAL))) by A13, A12, A11, A16, NAT_LAT:def 12;

take L9 ; :: thesis: ( L9 is \/-inheriting & not L9 is /\-inheriting )

now :: thesis: for X being Subset of L9 holds "\/" (X,L) in the carrier of L9

hence
L9 is \/-inheriting
; :: thesis: not L9 is /\-inheriting let X be Subset of L9; :: thesis: "\/" (X,L) in the carrier of L9

thus "\/" (X,L) in the carrier of L9 :: thesis: verum

end;thus "\/" (X,L) in the carrier of L9 :: thesis: verum

proof

0 in { r where r is Real : ( 0 <= r & r <= 1 ) }
;

then reconsider w = 0 as Element of L by A13, RCOMP_1:def 1;

A56: X is_less_than "\/" (X,L) by LATTICE3:def 21;

"\/" (X,L) in [.0,1.] by A13;

then "\/" (X,L) in { r where r is Real : ( 0 <= r & r <= 1 ) } by RCOMP_1:def 1;

then consider y being Real such that

A57: y = "\/" (X,L) and

0 <= y and

A58: y <= 1 ;

reconsider y = y as Real ;

assume A59: not "\/" (X,L) in the carrier of L9 ; :: thesis: contradiction

then not "\/" (X,L) in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } by A9, XBOOLE_0:def 3;

then A60: y <= 1 / 2 by A57, A58;

end;then reconsider w = 0 as Element of L by A13, RCOMP_1:def 1;

A56: X is_less_than "\/" (X,L) by LATTICE3:def 21;

"\/" (X,L) in [.0,1.] by A13;

then "\/" (X,L) in { r where r is Real : ( 0 <= r & r <= 1 ) } by RCOMP_1:def 1;

then consider y being Real such that

A57: y = "\/" (X,L) and

0 <= y and

A58: y <= 1 ;

reconsider y = y as Real ;

assume A59: not "\/" (X,L) in the carrier of L9 ; :: thesis: contradiction

then not "\/" (X,L) in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } by A9, XBOOLE_0:def 3;

then A60: y <= 1 / 2 by A57, A58;

now :: thesis: for z9 being object st z9 in X holds

z9 in {0}

then A63:
X c= {0}
;z9 in {0}

let z9 be object ; :: thesis: ( z9 in X implies z9 in {0} )

assume A61: z9 in X ; :: thesis: z9 in {0}

then reconsider z = z9 as Element of L9 ;

reconsider z = z as Element of L by A13, A14;

A62: z [= "\/" (X,L) by A56, A61;

reconsider z1 = z as Real ;

reconsider z1 = z1 as Real ;

min (z1,y) = minreal . (z1,("\/" (X,L))) by A57, REAL_LAT:def 1

.= (minreal || A) . [z,("\/" (X,L))] by A13, FUNCT_1:49

.= (minreal || A) . (z,("\/" (X,L)))

.= z "/\" ("\/" (X,L)) by A11, LATTICES:def 2

.= z1 by A62, LATTICES:4 ;

then z1 <= y by XXREAL_0:def 9;

then y + z1 <= (1 / 2) + y by A60, XREAL_1:7;

then for v being Real holds

( not z1 = v or not 1 / 2 < v or not v <= 1 ) by XREAL_1:6;

then not z1 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } ;

hence z9 in {0} by A9, XBOOLE_0:def 3; :: thesis: verum

end;assume A61: z9 in X ; :: thesis: z9 in {0}

then reconsider z = z9 as Element of L9 ;

reconsider z = z as Element of L by A13, A14;

A62: z [= "\/" (X,L) by A56, A61;

reconsider z1 = z as Real ;

reconsider z1 = z1 as Real ;

min (z1,y) = minreal . (z1,("\/" (X,L))) by A57, REAL_LAT:def 1

.= (minreal || A) . [z,("\/" (X,L))] by A13, FUNCT_1:49

.= (minreal || A) . (z,("\/" (X,L)))

.= z "/\" ("\/" (X,L)) by A11, LATTICES:def 2

.= z1 by A62, LATTICES:4 ;

then z1 <= y by XXREAL_0:def 9;

then y + z1 <= (1 / 2) + y by A60, XREAL_1:7;

then for v being Real holds

( not z1 = v or not 1 / 2 < v or not v <= 1 ) by XREAL_1:6;

then not z1 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } ;

hence z9 in {0} by A9, XBOOLE_0:def 3; :: thesis: verum

now :: thesis: contradictionend;

hence
contradiction
; :: thesis: verumper cases
( X = {} or X = {0} )
by A63, ZFMISC_1:33;

end;

suppose A64:
X = {}
; :: thesis: contradiction

q [= w by A64;

then X is_less_than w ;

then "\/" (X,L) = w by A65, LATTICE3:def 21;

then "\/" (X,L) in {0} by TARSKI:def 1;

hence contradiction by A9, A59, XBOOLE_0:def 3; :: thesis: verum

end;

A65: now :: thesis: for r1 being Element of L st X is_less_than r1 holds

w [= r1

for q being Element of L st q in X holds w [= r1

let r1 be Element of L; :: thesis: ( X is_less_than r1 implies w [= r1 )

assume X is_less_than r1 ; :: thesis: w [= r1

r1 in [.0,1.] by A13;

then r1 in { r where r is Real : ( 0 <= r & r <= 1 ) } by RCOMP_1:def 1;

then consider e being Real such that

A66: r1 = e and

A67: 0 <= e and

e <= 1 ;

reconsider e = e as Real ;

w "/\" r1 = (minreal || A) . (w,r1) by A11, LATTICES:def 2

.= minreal . [w,r1] by A13, FUNCT_1:49

.= minreal . (w,r1)

.= min (0,e) by A66, REAL_LAT:def 1

.= w by A67, XXREAL_0:def 9 ;

hence w [= r1 by LATTICES:4; :: thesis: verum

end;assume X is_less_than r1 ; :: thesis: w [= r1

r1 in [.0,1.] by A13;

then r1 in { r where r is Real : ( 0 <= r & r <= 1 ) } by RCOMP_1:def 1;

then consider e being Real such that

A66: r1 = e and

A67: 0 <= e and

e <= 1 ;

reconsider e = e as Real ;

w "/\" r1 = (minreal || A) . (w,r1) by A11, LATTICES:def 2

.= minreal . [w,r1] by A13, FUNCT_1:49

.= minreal . (w,r1)

.= min (0,e) by A66, REAL_LAT:def 1

.= w by A67, XXREAL_0:def 9 ;

hence w [= r1 by LATTICES:4; :: thesis: verum

q [= w by A64;

then X is_less_than w ;

then "\/" (X,L) = w by A65, LATTICE3:def 21;

then "\/" (X,L) in {0} by TARSKI:def 1;

hence contradiction by A9, A59, XBOOLE_0:def 3; :: thesis: verum

suppose
X = {0}
; :: thesis: contradiction

then
"\/" (X,L) = w
by LATTICE3:42;

then "\/" (X,L) in {0} by TARSKI:def 1;

hence contradiction by A9, A59, XBOOLE_0:def 3; :: thesis: verum

end;then "\/" (X,L) in {0} by TARSKI:def 1;

hence contradiction by A9, A59, XBOOLE_0:def 3; :: thesis: verum

now :: thesis: not for X being Subset of L9 holds "/\" (X,L) in the carrier of L9

hence
not L9 is /\-inheriting
; :: thesis: verum
1 / 2 in { x where x is Real : ( 0 <= x & x <= 1 ) }
;

then reconsider z = 1 / 2 as Element of L by A13, RCOMP_1:def 1;

set X = { x where x is Real : ( 1 / 2 < x & x <= 1 ) } ;

for y being Real holds

( not y = 1 / 2 or not 1 / 2 < y or not y <= 1 ) ;

then A68: ( not 1 / 2 in {0} & not 1 / 2 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } ) by TARSKI:def 1;

for x1 being object st x1 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } holds

x1 in B by A9, XBOOLE_0:def 3;

then reconsider X = { x where x is Real : ( 1 / 2 < x & x <= 1 ) } as Subset of L9 by TARSKI:def 3;

take X = X; :: thesis: not "/\" (X,L) in the carrier of L9

then "/\" (X,L) = jd by A69, LATTICE3:34;

hence not "/\" (X,L) in the carrier of L9 by A1, A68, XBOOLE_0:def 3; :: thesis: verum

end;then reconsider z = 1 / 2 as Element of L by A13, RCOMP_1:def 1;

set X = { x where x is Real : ( 1 / 2 < x & x <= 1 ) } ;

for y being Real holds

( not y = 1 / 2 or not 1 / 2 < y or not y <= 1 ) ;

then A68: ( not 1 / 2 in {0} & not 1 / 2 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } ) by TARSKI:def 1;

for x1 being object st x1 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } holds

x1 in B by A9, XBOOLE_0:def 3;

then reconsider X = { x where x is Real : ( 1 / 2 < x & x <= 1 ) } as Subset of L9 by TARSKI:def 3;

take X = X; :: thesis: not "/\" (X,L) in the carrier of L9

A69: now :: thesis: for b being Element of L st b is_less_than X holds

b [= z

b [= z

let b be Element of L; :: thesis: ( b is_less_than X implies b [= z )

b in A by A13;

then b in { r where r is Real : ( 0 <= r & r <= 1 ) } by RCOMP_1:def 1;

then consider b9 being Real such that

A70: b = b9 and

A71: 0 <= b9 and

A72: b9 <= 1 ;

reconsider b9 = b9 as Real ;

assume A73: b is_less_than X ; :: thesis: b [= z

A74: b9 <= 1 / 2

.= minreal . [b,z] by A13, FUNCT_1:49

.= minreal . (b,z)

.= min (b9,jd) by A70, REAL_LAT:def 1

.= b by A70, A74, XXREAL_0:def 9 ;

hence b [= z by LATTICES:4; :: thesis: verum

end;b in A by A13;

then b in { r where r is Real : ( 0 <= r & r <= 1 ) } by RCOMP_1:def 1;

then consider b9 being Real such that

A70: b = b9 and

A71: 0 <= b9 and

A72: b9 <= 1 ;

reconsider b9 = b9 as Real ;

assume A73: b is_less_than X ; :: thesis: b [= z

A74: b9 <= 1 / 2

proof

b "/\" z =
(minreal || A) . (b,z)
by A11, LATTICES:def 2
assume A75:
1 / 2 < b9
; :: thesis: contradiction

then b9 in X by A72;

then A76: b = "/\" (X,L) by A70, A73, LATTICE3:41;

(1 / 2) + b9 < b9 + b9 by A75, XREAL_1:6;

then A77: ((1 / 2) + b9) / 2 < (b9 + b9) / 2 by XREAL_1:74;

then (((1 / 2) + b9) / 2) + b9 <= b9 + 1 by A72, XREAL_1:7;

then A78: ((1 / 2) + b9) / 2 <= 1 by XREAL_1:6;

then ((1 / 2) + b9) / 2 in { r where r is Real : ( 0 <= r & r <= 1 ) } by A71;

then reconsider c = ((1 / 2) + b9) / 2 as Element of L by A13, RCOMP_1:def 1;

reconsider cc = c as Real ;

(1 / 2) + (1 / 2) < b9 + (1 / 2) by A75, XREAL_1:6;

then 1 / 2 < ((1 / 2) + b9) / 2 by XREAL_1:74;

then ((1 / 2) + b9) / 2 in X by A78;

then b [= c by A76, LATTICE3:38;

then b9 = b "/\" c by A70, LATTICES:4

.= (minreal || A) . (b,c) by A11, LATTICES:def 2

.= minreal . [b,c] by A13, FUNCT_1:49

.= minreal . (b,cc)

.= min (b9,(((1 / 2) + b9) / 2)) by A70, REAL_LAT:def 1 ;

hence contradiction by A77, XXREAL_0:def 9; :: thesis: verum

end;then b9 in X by A72;

then A76: b = "/\" (X,L) by A70, A73, LATTICE3:41;

(1 / 2) + b9 < b9 + b9 by A75, XREAL_1:6;

then A77: ((1 / 2) + b9) / 2 < (b9 + b9) / 2 by XREAL_1:74;

then (((1 / 2) + b9) / 2) + b9 <= b9 + 1 by A72, XREAL_1:7;

then A78: ((1 / 2) + b9) / 2 <= 1 by XREAL_1:6;

then ((1 / 2) + b9) / 2 in { r where r is Real : ( 0 <= r & r <= 1 ) } by A71;

then reconsider c = ((1 / 2) + b9) / 2 as Element of L by A13, RCOMP_1:def 1;

reconsider cc = c as Real ;

(1 / 2) + (1 / 2) < b9 + (1 / 2) by A75, XREAL_1:6;

then 1 / 2 < ((1 / 2) + b9) / 2 by XREAL_1:74;

then ((1 / 2) + b9) / 2 in X by A78;

then b [= c by A76, LATTICE3:38;

then b9 = b "/\" c by A70, LATTICES:4

.= (minreal || A) . (b,c) by A11, LATTICES:def 2

.= minreal . [b,c] by A13, FUNCT_1:49

.= minreal . (b,cc)

.= min (b9,(((1 / 2) + b9) / 2)) by A70, REAL_LAT:def 1 ;

hence contradiction by A77, XXREAL_0:def 9; :: thesis: verum

.= minreal . [b,z] by A13, FUNCT_1:49

.= minreal . (b,z)

.= min (b9,jd) by A70, REAL_LAT:def 1

.= b by A70, A74, XXREAL_0:def 9 ;

hence b [= z by LATTICES:4; :: thesis: verum

now :: thesis: for q being Element of L st q in X holds

z [= q

then
z is_less_than X
;z [= q

let q be Element of L; :: thesis: ( q in X implies z [= q )

assume q in X ; :: thesis: z [= q

then A79: ex y being Real st

( q = y & 1 / 2 < y & y <= 1 ) ;

q in A by A13;

then q in { r where r is Real : ( 0 <= r & r <= 1 ) } by RCOMP_1:def 1;

then consider q9 being Real such that

A80: q = q9 and

0 <= q9 and

q9 <= 1 ;

reconsider q9 = q9 as Real ;

z "/\" q = (minreal || A) . (z,q) by A11, LATTICES:def 2

.= minreal . [z,q] by A13, FUNCT_1:49

.= minreal . (z,q)

.= min (jd,q9) by A80, REAL_LAT:def 1

.= z by A80, A79, XXREAL_0:def 9 ;

hence z [= q by LATTICES:4; :: thesis: verum

end;assume q in X ; :: thesis: z [= q

then A79: ex y being Real st

( q = y & 1 / 2 < y & y <= 1 ) ;

q in A by A13;

then q in { r where r is Real : ( 0 <= r & r <= 1 ) } by RCOMP_1:def 1;

then consider q9 being Real such that

A80: q = q9 and

0 <= q9 and

q9 <= 1 ;

reconsider q9 = q9 as Real ;

z "/\" q = (minreal || A) . (z,q) by A11, LATTICES:def 2

.= minreal . [z,q] by A13, FUNCT_1:49

.= minreal . (z,q)

.= min (jd,q9) by A80, REAL_LAT:def 1

.= z by A80, A79, XXREAL_0:def 9 ;

hence z [= q by LATTICES:4; :: thesis: verum

then "/\" (X,L) = jd by A69, LATTICE3:34;

hence not "/\" (X,L) in the carrier of L9 by A1, A68, XBOOLE_0:def 3; :: thesis: verum