:: PCS_0 semantic presentation

begin

reconsider z = 0 , j = 1 as Element of {0,1} by TARSKI:def_2;

definition
let R1, R2 be set ;
let R be Relation of R1,R2;
:: original: field
redefine func field R ->  Subset of (R1 \/ R2);
coherence
 field R is Subset of (R1 \/ R2) by RELSET_1:8;
end;

definition
let R1, R2, S1, S2 be set ;
let R be Relation of R1,R2;
let S be Relation of S1,S2;
:: original: \/
redefine funcR \/ S ->  Relation of (R1 \/ S1),(R2 \/ S2);
coherence
 R \/ S is Relation of (R1 \/ S1),(R2 \/ S2) by ZFMISC_1:119;
end;

registration
let R1, S1 be set ;
let R be total Relation of R1;
let S be total Relation of S1;
clusterR \/ S ->  total for Relation of (R1 \/ S1);
coherence
 for b1 being Relation of (R1 \/ S1) st b1 = R \/ S holds
b1 is total
proof
dom (R \/ S) = (dom R) \/ (dom S) by RELAT_1:1
.= R1 \/ (dom S) by PARTFUN1:def_2
.= R1 \/ S1 by PARTFUN1:def_2 ;
hence  for b1 being Relation of (R1 \/ S1) st b1 = R \/ S holds
b1 is total by PARTFUN1:def_2; ::_thesis: verum
end;
end;

registration
let R1, S1 be set ;
let R be reflexive Relation of R1;
let S be reflexive Relation of S1;
clusterR \/ S ->  reflexive for Relation of (R1 \/ S1);
coherence
 for b1 being Relation of (R1 \/ S1) st b1 = R \/ S holds
b1 is reflexive ;
end;

registration
let R1, S1 be set ;
let R be symmetric Relation of R1;
let S be symmetric Relation of S1;
clusterR \/ S ->  symmetric for Relation of (R1 \/ S1);
coherence
 for b1 being Relation of (R1 \/ S1) st b1 = R \/ S holds
b1 is symmetric ;
end;

Lm1: now__::_thesis:_for_R1,_S1_being_set_
for_R_being_Relation_of_R1
for_S_being_Relation_of_S1_st_R1_misses_S1_holds_
for_q1,_q2_being_set_st_[q1,q2]_in_R_\/_S_&_q2_in_S1_holds_
(_[q1,q2]_in_S_&_q1_in_S1_)
let R1, S1 be set ; ::_thesis: for R being Relation of R1
for S being Relation of S1 st R1 misses S1 holds
for q1, q2 being set st [q1,q2] in R \/ S & q2 in S1 holds
( [q1,q2] in S & q1 in S1 )
let R be Relation of R1; ::_thesis: for S being Relation of S1 st R1 misses S1 holds
for q1, q2 being set st [q1,q2] in R \/ S & q2 in S1 holds
( [q1,q2] in S & q1 in S1 )
let S be Relation of S1; ::_thesis: ( R1 misses S1 implies for q1, q2 being set st [q1,q2] in R \/ S & q2 in S1 holds
( [q1,q2] in S & q1 in S1 ) )
assume A1: R1 misses S1 ; ::_thesis: for q1, q2 being set st [q1,q2] in R \/ S & q2 in S1 holds
( [q1,q2] in S & q1 in S1 )
let q1, q2 be set ; ::_thesis: ( [q1,q2] in R \/ S & q2 in S1 implies ( [q1,q2] in S & q1 in S1 ) )
assume that
A2: [q1,q2] in R \/ S and
A3: q2 in S1 ; ::_thesis: ( [q1,q2] in S & q1 in S1 )
A4: ( [q1,q2] in R or [q1,q2] in S ) by A2, XBOOLE_0:def_3;
now__::_thesis:_not_[q1,q2]_in_R
assume  [q1,q2] in R ; ::_thesis: contradiction
then  q2 in R1 by ZFMISC_1:87;
hence  contradiction by A1, A3, XBOOLE_0:3; ::_thesis: verum
end;
hence  ( [q1,q2] in S & q1 in S1 ) by A4, ZFMISC_1:87; ::_thesis: verum
end;

theorem Th1: :: PCS_0:1
for R1, S1 being set
for R being transitive Relation of R1
for S being transitive Relation of S1 st R1 misses S1 holds
R \/ S is transitive
proof
let R1, S1 be set ; ::_thesis: for R being transitive Relation of R1
for S being transitive Relation of S1 st R1 misses S1 holds
R \/ S is transitive
let R be transitive Relation of R1; ::_thesis: for S being transitive Relation of S1 st R1 misses S1 holds
R \/ S is transitive
let S be transitive Relation of S1; ::_thesis: ( R1 misses S1 implies R \/ S is transitive )
assume A1: R1 misses S1 ; ::_thesis: R \/ S is transitive
let p1, p2, p3 be set ; :: according to RELAT_2:def_8,RELAT_2:def_16 ::_thesis: ( not p1 in field (R \/ S) or not p2 in field (R \/ S) or not p3 in field (R \/ S) or not [^,^] in R \/ S or not [^,^] in R \/ S or [^,^] in R \/ S )
set RS = R \/ S;
set D = field (R \/ S);
assume that
 p1 in field (R \/ S) and
 p2 in field (R \/ S) and
A2: p3 in field (R \/ S) and
A3: [p1,p2] in R \/ S and
A4: [p2,p3] in R \/ S ; ::_thesis: [^,^] in R \/ S
percases ( p3 in R1 or p3 in S1 ) by A2, XBOOLE_0:def_3;
supposeA5: p3 in R1 ; ::_thesis: [^,^] in R \/ S
then  p2 in R1 by A1, A4, Lm1;
then A6: [p1,p2] in R by A1, A3, Lm1;
 [p2,p3] in R by A1, A4, A5, Lm1;
then  [p1,p3] in R by A6, RELAT_2:31;
hence  [^,^] in R \/ S by XBOOLE_0:def_3; ::_thesis: verum
end;
supposeA7: p3 in S1 ; ::_thesis: [^,^] in R \/ S
then  p2 in S1 by A1, A4, Lm1;
then A8: [p1,p2] in S by A1, A3, Lm1;
 [p2,p3] in S by A1, A4, A7, Lm1;
then  [p1,p3] in S by A8, RELAT_2:31;
hence  [^,^] in R \/ S by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;

definition
let I be non empty set ;
let C be 1-sorted-yielding ManySortedSet of I;
redefine func Carrier C means :Def1: :: PCS_0:def 1
 for i being Element of I holds it . i = the carrier of (C . i);
compatibility
 for b1 being set holds
( b1 = Carrier C iff for i being Element of I holds b1 . i = the carrier of (C . i) )
proof
let X be ManySortedSet of I; ::_thesis: ( X = Carrier C iff for i being Element of I holds X . i = the carrier of (C . i) )
thus  ( X = Carrier C implies for i being Element of I holds X . i = the carrier of (C . i) ) ::_thesis: ( ( for i being Element of I holds X . i = the carrier of (C . i) ) implies X = Carrier C )
proof
assume A1: X = Carrier C ; ::_thesis: for i being Element of I holds X . i = the carrier of (C . i)
let i be Element of I; ::_thesis: X . i = the carrier of (C . i)
 ex P being 1-sorted st
( P = C . i & X . i = the carrier of P ) by A1, PRALG_1:def_13;
hence  X . i = the carrier of (C . i) ; ::_thesis: verum
end;
assume A2: for i being Element of I holds X . i = the carrier of (C . i) ; ::_thesis: X = Carrier C
 for i being set st i in I holds
ex P being 1-sorted st
( P = C . i & X . i = the carrier of P )
proof
let i be set ; ::_thesis: ( i in I implies ex P being 1-sorted st
( P = C . i & X . i = the carrier of P ) )
assume  i in I ; ::_thesis: ex P being 1-sorted st
( P = C . i & X . i = the carrier of P )
then reconsider i = i as Element of I ;
take  C . i ; ::_thesis: ( C . i = C . i & X . i = the carrier of (C . i) )
thus  ( C . i = C . i & X . i = the carrier of (C . i) ) by A2; ::_thesis: verum
end;
hence  X = Carrier C by PRALG_1:def_13; ::_thesis: verum
end;
end;

:: deftheorem Def1 defines Carrier PCS_0:def_1_:_
 for I being non empty set
for C being 1-sorted-yielding ManySortedSet of I
for b3 being set holds
( b3 = Carrier C iff for i being Element of I holds b3 . i = the carrier of (C . i) );

definition
let R1, R2, S1, S2 be set ;
let R be Relation of R1,R2;
let S be Relation of S1,S2;
defpred S1[ set , set ] means  ex r1, s1, r2, s2 being set st
( $1 = [r1,s1] & $2 = [r2,s2] & r1 in R1 & s1 in S1 & r2 in R2 & s2 in S2 & ( [r1,r2] in R or [s1,s2] in S ) );
func[^R,S^] ->  Relation of [:R1,S1:],[:R2,S2:] means :Def2: :: PCS_0:def 2
 for x, y being set holds
( [x,y] in it iff ex r1, s1, r2, s2 being set st
( x = [r1,s1] & y = [r2,s2] & r1 in R1 & s1 in S1 & r2 in R2 & s2 in S2 & ( [r1,r2] in R or [s1,s2] in S ) ) );
existence
 ex b1 being Relation of [:R1,S1:],[:R2,S2:] st
for x, y being set holds
( [x,y] in b1 iff ex r1, s1, r2, s2 being set st
( x = [r1,s1] & y = [r2,s2] & r1 in R1 & s1 in S1 & r2 in R2 & s2 in S2 & ( [r1,r2] in R or [s1,s2] in S ) ) )
proof
consider Q being Relation of [:R1,S1:],[:R2,S2:] such that
A1: for x, y being set holds
( [x,y] in Q iff ( x in [:R1,S1:] & y in [:R2,S2:] & S1[x,y] ) ) from RELSET_1:sch_1();
take  Q ; ::_thesis: for x, y being set holds
( [x,y] in Q iff ex r1, s1, r2, s2 being set st
( x = [r1,s1] & y = [r2,s2] & r1 in R1 & s1 in S1 & r2 in R2 & s2 in S2 & ( [r1,r2] in R or [s1,s2] in S ) ) )
let x, y be set ; ::_thesis: ( [x,y] in Q iff ex r1, s1, r2, s2 being set st
( x = [r1,s1] & y = [r2,s2] & r1 in R1 & s1 in S1 & r2 in R2 & s2 in S2 & ( [r1,r2] in R or [s1,s2] in S ) ) )
thus  ( [x,y] in Q implies S1[x,y] ) by A1; ::_thesis: ( ex r1, s1, r2, s2 being set st
( x = [r1,s1] & y = [r2,s2] & r1 in R1 & s1 in S1 & r2 in R2 & s2 in S2 & ( [r1,r2] in R or [s1,s2] in S ) ) implies [x,y] in Q )
given r1, s1, r2, s2 being set  such that A2: x = [r1,s1] and
A3: y = [r2,s2] and
A4: r1 in R1 and
A5: s1 in S1 and
A6: r2 in R2 and
A7: s2 in S2 and
A8: ( [r1,r2] in R or [s1,s2] in S ) ; ::_thesis: [x,y] in Q
A9: x in [:R1,S1:] by A2, A4, A5, ZFMISC_1:87;
 y in [:R2,S2:] by A3, A6, A7, ZFMISC_1:87;
hence  [x,y] in Q by A1, A2, A3, A4, A5, A6, A7, A8, A9; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Relation of [:R1,S1:],[:R2,S2:] st ( for x, y being set holds
( [x,y] in b1 iff ex r1, s1, r2, s2 being set st
( x = [r1,s1] & y = [r2,s2] & r1 in R1 & s1 in S1 & r2 in R2 & s2 in S2 & ( [r1,r2] in R or [s1,s2] in S ) ) ) ) & ( for x, y being set holds
( [x,y] in b2 iff ex r1, s1, r2, s2 being set st
( x = [r1,s1] & y = [r2,s2] & r1 in R1 & s1 in S1 & r2 in R2 & s2 in S2 & ( [r1,r2] in R or [s1,s2] in S ) ) ) ) holds
b1 = b2
proof
let A, B be Relation of [:R1,S1:],[:R2,S2:]; ::_thesis: ( ( for x, y being set holds
( [x,y] in A iff ex r1, s1, r2, s2 being set st
( x = [r1,s1] & y = [r2,s2] & r1 in R1 & s1 in S1 & r2 in R2 & s2 in S2 & ( [r1,r2] in R or [s1,s2] in S ) ) ) ) & ( for x, y being set holds
( [x,y] in B iff ex r1, s1, r2, s2 being set st
( x = [r1,s1] & y = [r2,s2] & r1 in R1 & s1 in S1 & r2 in R2 & s2 in S2 & ( [r1,r2] in R or [s1,s2] in S ) ) ) ) implies A = B )
assume that
A10: for x, y being set holds
( [x,y] in A iff S1[x,y] ) and
A11: for x, y being set holds
( [x,y] in B iff S1[x,y] ) ; ::_thesis: A = B
thus  A = B from RELAT_1:sch_2(A10, A11); ::_thesis: verum
end;
end;

:: deftheorem Def2 defines [^ PCS_0:def_2_:_
 for R1, R2, S1, S2 being set
for R being Relation of R1,R2
for S being Relation of S1,S2
for b7 being Relation of [:R1,S1:],[:R2,S2:] holds
( b7 = [^R,S^] iff for x, y being set holds
( [x,y] in b7 iff ex r1, s1, r2, s2 being set st
( x = [r1,s1] & y = [r2,s2] & r1 in R1 & s1 in S1 & r2 in R2 & s2 in S2 & ( [r1,r2] in R or [s1,s2] in S ) ) ) );

definition
let R1, R2, S1, S2 be non empty set ;
let R be Relation of R1,R2;
let S be Relation of S1,S2;
redefine func [^R,S^] means :Def3: :: PCS_0:def 3
 for r1 being Element of R1
for r2 being Element of R2
for s1 being Element of S1
for s2 being Element of S2 holds
( [[r1,s1],[r2,s2]] in it iff ( [r1,r2] in R or [s1,s2] in S ) );
compatibility
 for b1 being Relation of [:R1,S1:],[:R2,S2:] holds
( b1 = [^R,S^] iff for r1 being Element of R1
for r2 being Element of R2
for s1 being Element of S1
for s2 being Element of S2 holds
( [[r1,s1],[r2,s2]] in b1 iff ( [r1,r2] in R or [s1,s2] in S ) ) )
proof
let f be Relation of [:R1,S1:],[:R2,S2:]; ::_thesis: ( f = [^R,S^] iff for r1 being Element of R1
for r2 being Element of R2
for s1 being Element of S1
for s2 being Element of S2 holds
( [[r1,s1],[r2,s2]] in f iff ( [r1,r2] in R or [s1,s2] in S ) ) )
thus  ( f = [^R,S^] implies for r1 being Element of R1
for r2 being Element of R2
for s1 being Element of S1
for s2 being Element of S2 holds
( [[r1,s1],[r2,s2]] in f iff ( [r1,r2] in R or [s1,s2] in S ) ) ) ::_thesis: ( ( for r1 being Element of R1
for r2 being Element of R2
for s1 being Element of S1
for s2 being Element of S2 holds
( [[r1,s1],[r2,s2]] in f iff ( [r1,r2] in R or [s1,s2] in S ) ) ) implies f = [^R,S^] )
proof
assume A1: f = [^R,S^] ; ::_thesis: for r1 being Element of R1
for r2 being Element of R2
for s1 being Element of S1
for s2 being Element of S2 holds
( [[r1,s1],[r2,s2]] in f iff ( [r1,r2] in R or [s1,s2] in S ) )
let r1 be Element of R1; ::_thesis: for r2 being Element of R2
for s1 being Element of S1
for s2 being Element of S2 holds
( [[r1,s1],[r2,s2]] in f iff ( [r1,r2] in R or [s1,s2] in S ) )
let r2 be Element of R2; ::_thesis: for s1 being Element of S1
for s2 being Element of S2 holds
( [[r1,s1],[r2,s2]] in f iff ( [r1,r2] in R or [s1,s2] in S ) )
let s1 be Element of S1; ::_thesis: for s2 being Element of S2 holds
( [[r1,s1],[r2,s2]] in f iff ( [r1,r2] in R or [s1,s2] in S ) )
let s2 be Element of S2; ::_thesis: ( [[r1,s1],[r2,s2]] in f iff ( [r1,r2] in R or [s1,s2] in S ) )
hereby  ::_thesis: ( ( [r1,r2] in R or [s1,s2] in S ) implies [[r1,s1],[r2,s2]] in f )
assume  [[r1,s1],[r2,s2]] in f ; ::_thesis: ( [r1,r2] in R or [s1,s2] in S )
then consider r19, s19, r29, s29 being set  such that
A2: [r1,s1] = [r19,s19] and
A3: [r2,s2] = [r29,s29] and
 r19 in R1 and
 s19 in S1 and
 r29 in R2 and
 s29 in S2 and
A4: ( [r19,r29] in R or [s19,s29] in S ) by A1, Def2;
A5: r1 = r19 by A2, XTUPLE_0:1;
 s1 = s19 by A2, XTUPLE_0:1;
hence  ( [r1,r2] in R or [s1,s2] in S ) by A3, A4, A5, XTUPLE_0:1; ::_thesis: verum
end;
thus  ( ( [r1,r2] in R or [s1,s2] in S ) implies [[r1,s1],[r2,s2]] in f ) by A1, Def2; ::_thesis: verum
end;
assume A6: for r1 being Element of R1
for r2 being Element of R2
for s1 being Element of S1
for s2 being Element of S2 holds
( [[r1,s1],[r2,s2]] in f iff ( [r1,r2] in R or [s1,s2] in S ) ) ; ::_thesis: f = [^R,S^]
 for x, y being set holds
( [x,y] in f iff [x,y] in [^R,S^] )
proof
let x, y be set ; ::_thesis: ( [x,y] in f iff [x,y] in [^R,S^] )
thus  ( [x,y] in f implies [x,y] in [^R,S^] ) ::_thesis: ( [x,y] in [^R,S^] implies [x,y] in f )
proof
assume A7: [x,y] in f ; ::_thesis: [x,y] in [^R,S^]
then  x in dom f by XTUPLE_0:def_12;
then consider x1, x2 being set  such that
A8: x1 in R1 and
A9: x2 in S1 and
A10: x = [x1,x2] by ZFMISC_1:def_2;
 y in rng f by A7, XTUPLE_0:def_13;
then consider y1, y2 being set  such that
A11: y1 in R2 and
A12: y2 in S2 and
A13: y = [y1,y2] by ZFMISC_1:def_2;
 ( [x1,y1] in R or [x2,y2] in S ) by A6, A7, A8, A9, A10, A11, A12, A13;
hence  [x,y] in [^R,S^] by A8, A9, A10, A11, A12, A13, Def2; ::_thesis: verum
end;
assume  [x,y] in [^R,S^] ; ::_thesis: [x,y] in f
then  ex r1, s1, r2, s2 being set st
( x = [r1,s1] & y = [r2,s2] & r1 in R1 & s1 in S1 & r2 in R2 & s2 in S2 & ( [r1,r2] in R or [s1,s2] in S ) ) by Def2;
hence  [x,y] in f by A6; ::_thesis: verum
end;
hence  f = [^R,S^] by RELAT_1:def_2; ::_thesis: verum
end;
end;

:: deftheorem Def3 defines [^ PCS_0:def_3_:_
 for R1, R2, S1, S2 being non empty set
for R being Relation of R1,R2
for S being Relation of S1,S2
for b7 being Relation of [:R1,S1:],[:R2,S2:] holds
( b7 = [^R,S^] iff for r1 being Element of R1
for r2 being Element of R2
for s1 being Element of S1
for s2 being Element of S2 holds
( [[r1,s1],[r2,s2]] in b7 iff ( [r1,r2] in R or [s1,s2] in S ) ) );

registration
let R1, S1 be set ;
let R be total Relation of R1;
let S be total Relation of S1;
cluster[^R,S^] ->  total ;
coherence
 [^R,S^] is total
proof
thus  dom [^R,S^] c= [:R1,S1:] ; :: according to PARTFUN1:def_2,XBOOLE_0:def_10 ::_thesis: [:R1,S1:] c= dom [^R,S^]
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in [:R1,S1:] or z in dom [^R,S^] )
assume  z in [:R1,S1:] ; ::_thesis: z in dom [^R,S^]
then consider x, y being set  such that
A1: x in R1 and
A2: y in S1 and
A3: z = [x,y] by ZFMISC_1:def_2;
 dom R = R1 by PARTFUN1:def_2;
then consider a being set  such that
A4: [x,a] in R by A1, XTUPLE_0:def_12;
 dom S = S1 by PARTFUN1:def_2;
then consider b being set  such that
A5: [y,b] in S by A2, XTUPLE_0:def_12;
A6: a in R1 by A4, ZFMISC_1:87;
 b in S1 by A5, ZFMISC_1:87;
then  [[x,y],[a,b]] in [^R,S^] by A1, A2, A4, A6, Def2;
hence  z in dom [^R,S^] by A3, XTUPLE_0:def_12; ::_thesis: verum
end;
end;

registration
let R1, S1 be set ;
let R be reflexive Relation of R1;
let S be reflexive Relation of S1;
cluster[^R,S^] ->  reflexive ;
coherence
 [^R,S^] is reflexive
proof
let x be set ; :: according to RELAT_2:def_1,RELAT_2:def_9 ::_thesis: ( not x in field [^R,S^] or [^,^] in [^R,S^] )
assume A1: x in field [^R,S^] ; ::_thesis: [^,^] in [^R,S^]
A2: R is_reflexive_in field R by RELAT_2:def_9;
A3: S is_reflexive_in field S by RELAT_2:def_9;
percases ( x in dom [^R,S^] or x in rng [^R,S^] ) by A1, XBOOLE_0:def_3;
suppose x in dom [^R,S^] ; ::_thesis: [^,^] in [^R,S^]
then consider y being set  such that
A4: [x,y] in [^R,S^] by XTUPLE_0:def_12;
consider p, q, s, t being set  such that
A5: x = [p,q] and
 y = [s,t] and
A6: p in R1 and
A7: q in S1 and
 s in R1 and
 t in S1 and
A8: ( [p,s] in R or [q,t] in S ) by A4, Def2;
percases ( [p,s] in R or [q,t] in S ) by A8;
suppose [p,s] in R ; ::_thesis: [^,^] in [^R,S^]
then  p in field R by RELAT_1:15;
then  [p,p] in R by A2, RELAT_2:def_1;
hence  [^,^] in [^R,S^] by A5, A6, A7, Def2; ::_thesis: verum
end;
suppose [q,t] in S ; ::_thesis: [^,^] in [^R,S^]
then  q in field S by RELAT_1:15;
then  [q,q] in S by A3, RELAT_2:def_1;
hence  [^,^] in [^R,S^] by A5, A6, A7, Def2; ::_thesis: verum
end;
end;
end;
suppose x in rng [^R,S^] ; ::_thesis: [^,^] in [^R,S^]
then consider y being set  such that
A9: [y,x] in [^R,S^] by XTUPLE_0:def_13;
consider p, q, s, t being set  such that
 y = [p,q] and
A10: x = [s,t] and
 p in R1 and
 q in S1 and
A11: s in R1 and
A12: t in S1 and
A13: ( [p,s] in R or [q,t] in S ) by A9, Def2;
percases ( [p,s] in R or [q,t] in S ) by A13;
suppose [p,s] in R ; ::_thesis: [^,^] in [^R,S^]
then  s in field R by RELAT_1:15;
then  [s,s] in R by A2, RELAT_2:def_1;
hence  [^,^] in [^R,S^] by A10, A11, A12, Def2; ::_thesis: verum
end;
suppose [q,t] in S ; ::_thesis: [^,^] in [^R,S^]
then  t in field S by RELAT_1:15;
then  [t,t] in S by A3, RELAT_2:def_1;
hence  [^,^] in [^R,S^] by A10, A11, A12, Def2; ::_thesis: verum
end;
end;
end;
end;
end;
end;

registration
let R1, S1 be set ;
let R be Relation of R1;
let S be total reflexive Relation of S1;
cluster[^R,S^] ->  reflexive ;
coherence
 [^R,S^] is reflexive
proof
let x be set ; :: according to RELAT_2:def_1,RELAT_2:def_9 ::_thesis: ( not x in field [^R,S^] or [^,^] in [^R,S^] )
assume  x in field [^R,S^] ; ::_thesis: [^,^] in [^R,S^]
then consider x1, x2 being set  such that
A1: x1 in R1 and
A2: x2 in S1 and
A3: x = [x1,x2] by ZFMISC_1:def_2;
 S1 = field S by ORDERS_1:12;
then  S is_reflexive_in S1 by RELAT_2:def_9;
then  [x2,x2] in S by A2, RELAT_2:def_1;
hence  [^,^] in [^R,S^] by A1, A2, A3, Def3; ::_thesis: verum
end;
end;

registration
let R1, S1 be set ;
let R be total reflexive Relation of R1;
let S be Relation of S1;
cluster[^R,S^] ->  reflexive ;
coherence
 [^R,S^] is reflexive
proof
let x be set ; :: according to RELAT_2:def_1,RELAT_2:def_9 ::_thesis: ( not x in field [^R,S^] or [^,^] in [^R,S^] )
assume  x in field [^R,S^] ; ::_thesis: [^,^] in [^R,S^]
then consider x1, x2 being set  such that
A1: x1 in R1 and
A2: x2 in S1 and
A3: x = [x1,x2] by ZFMISC_1:def_2;
 R1 = field R by ORDERS_1:12;
then  R is_reflexive_in R1 by RELAT_2:def_9;
then  [x1,x1] in R by A1, RELAT_2:def_1;
hence  [^,^] in [^R,S^] by A1, A2, A3, Def3; ::_thesis: verum
end;
end;

registration
let R1, S1 be set ;
let R be symmetric Relation of R1;
let S be symmetric Relation of S1;
cluster[^R,S^] ->  symmetric ;
coherence
 [^R,S^] is symmetric
proof
let x, y be set ; :: according to RELAT_2:def_3,RELAT_2:def_11 ::_thesis: ( not x in field [^R,S^] or not y in field [^R,S^] or not [^,^] in [^R,S^] or [^,^] in [^R,S^] )
assume that
 x in field [^R,S^] and
 y in field [^R,S^] ; ::_thesis: ( not [^,^] in [^R,S^] or [^,^] in [^R,S^] )
assume  [x,y] in [^R,S^] ; ::_thesis: [^,^] in [^R,S^]
then consider p, q, s, t being set  such that
A1: x = [p,q] and
A2: y = [s,t] and
A3: p in R1 and
A4: q in S1 and
A5: s in R1 and
A6: t in S1 and
A7: ( [p,s] in R or [q,t] in S ) by Def2;
A8: R is_symmetric_in field R by RELAT_2:def_11;
A9: S is_symmetric_in field S by RELAT_2:def_11;
percases ( [p,s] in R or [q,t] in S ) by A7;
supposeA10: [p,s] in R ; ::_thesis: [^,^] in [^R,S^]
then A11: p in field R by RELAT_1:15;
 s in field R by A10, RELAT_1:15;
then  [s,p] in R by A8, A10, A11, RELAT_2:def_3;
hence  [^,^] in [^R,S^] by A1, A2, A3, A4, A5, A6, Def2; ::_thesis: verum
end;
supposeA12: [q,t] in S ; ::_thesis: [^,^] in [^R,S^]
then A13: q in field S by RELAT_1:15;
 t in field S by A12, RELAT_1:15;
then  [t,q] in S by A9, A12, A13, RELAT_2:def_3;
hence  [^,^] in [^R,S^] by A1, A2, A3, A4, A5, A6, Def2; ::_thesis: verum
end;
end;
end;
end;

begin

registration
cluster empty  ->  total for RelStr ;
coherence
 for b1 being RelStr st b1 is empty holds
b1 is total
proof
let P be RelStr ; ::_thesis: ( P is empty implies P is total )
assume  the carrier of P is empty ; :: according to STRUCT_0:def_1 ::_thesis: P is total
hence  dom the InternalRel of P = the carrier of P ; :: according to PARTFUN1:def_2,ORDERS_2:def_1 ::_thesis: verum
end;
end;

definition
let R be Relation;
attrR is transitive-yielding  means :Def4: :: PCS_0:def 4
 for S being RelStr st S in rng R holds
S is transitive ;
end;

:: deftheorem Def4 defines transitive-yielding PCS_0:def_4_:_
 for R being Relation holds
( R is transitive-yielding iff for S being RelStr st S in rng R holds
S is transitive );

registration
cluster Relation-like Poset-yielding  ->  transitive-yielding for set ;
coherence
 for b1 being Relation st b1 is Poset-yielding holds
b1 is transitive-yielding
proof
let R be Relation; ::_thesis: ( R is Poset-yielding implies R is transitive-yielding )
assume A1: R is Poset-yielding ; ::_thesis: R is transitive-yielding
let S be RelStr ; :: according to PCS_0:def_4 ::_thesis: ( S in rng R implies S is transitive )
thus  ( S in rng R implies S is transitive ) by A1, YELLOW16:def_5; ::_thesis: verum
end;
end;

registration
cluster Relation-like Function-like Poset-yielding for set ;
existence
 ex b1 being Function st b1 is Poset-yielding
proof
set f = the Poset-yielding ManySortedSet of 0 ;
take  the Poset-yielding ManySortedSet of 0 ; ::_thesis: the Poset-yielding ManySortedSet of 0 is Poset-yielding
thus  the Poset-yielding ManySortedSet of 0 is Poset-yielding ; ::_thesis: verum
end;
end;

registration
let I be set ;
cluster Relation-like I -defined Function-like total Poset-yielding for set ;
existence
 ex b1 being ManySortedSet of I st b1 is Poset-yielding
proof
set f = the Poset-yielding ManySortedSet of I;
take  the Poset-yielding ManySortedSet of I ; ::_thesis: the Poset-yielding ManySortedSet of I is Poset-yielding
thus  the Poset-yielding ManySortedSet of I is Poset-yielding ; ::_thesis: verum
end;
end;

definition
let I be set ;
let C be RelStr-yielding ManySortedSet of I;
func pcs-InternalRels C ->  ManySortedSet of I means :Def5: :: PCS_0:def 5
 for i being set st i in I holds
ex P being RelStr st
( P = C . i & it . i = the InternalRel of P );
existence
 ex b1 being ManySortedSet of I st
for i being set st i in I holds
ex P being RelStr st
( P = C . i & b1 . i = the InternalRel of P )
proof
defpred S1[ set , set ] means  ex R being RelStr st
( R = C . $1 & $2 = the InternalRel of R );
A1: for i being set st i in I holds
ex j being set st S1[i,j]
proof
let i be set ; ::_thesis: ( i in I implies ex j being set st S1[i,j] )
assume A2: i in I ; ::_thesis: ex j being set st S1[i,j]
then reconsider I = I as non empty set ;
reconsider B = C as RelStr-yielding ManySortedSet of I ;
reconsider i9 = i as Element of I by A2;
take  the InternalRel of (B . i9) ; ::_thesis: S1[i, the InternalRel of (B . i9)]
take  B . i9 ; ::_thesis: ( B . i9 = C . i & the InternalRel of (B . i9) = the InternalRel of (B . i9) )
thus  ( B . i9 = C . i & the InternalRel of (B . i9) = the InternalRel of (B . i9) ) ; ::_thesis: verum
end;
consider M being Function such that
A3: dom M = I and
A4: for i being set st i in I holds
S1[i,M . i] from CLASSES1:sch_1(A1);
 M is ManySortedSet of I by A3, PARTFUN1:def_2, RELAT_1:def_18;
hence  ex b1 being ManySortedSet of I st
for i being set st i in I holds
ex P being RelStr st
( P = C . i & b1 . i = the InternalRel of P ) by A4; ::_thesis: verum
end;
uniqueness
 for b1, b2 being ManySortedSet of I st ( for i being set st i in I holds
ex P being RelStr st
( P = C . i & b1 . i = the InternalRel of P ) ) & ( for i being set st i in I holds
ex P being RelStr st
( P = C . i & b2 . i = the InternalRel of P ) ) holds
b1 = b2
proof
let X, Y be ManySortedSet of I; ::_thesis: ( ( for i being set st i in I holds
ex P being RelStr st
( P = C . i & X . i = the InternalRel of P ) ) & ( for i being set st i in I holds
ex P being RelStr st
( P = C . i & Y . i = the InternalRel of P ) ) implies X = Y )
assume that
A5: for j being set st j in I holds
ex R being RelStr st
( R = C . j & X . j = the InternalRel of R ) and
A6: for j being set st j in I holds
ex R being RelStr st
( R = C . j & Y . j = the InternalRel of R ) ; ::_thesis: X = Y
 for i being set st i in I holds
X . i = Y . i
proof
let i be set ; ::_thesis: ( i in I implies X . i = Y . i )
assume A7: i in I ; ::_thesis: X . i = Y . i
then A8: ex R being RelStr st
( R = C . i & X . i = the InternalRel of R ) by A5;
 ex R being RelStr st
( R = C . i & Y . i = the InternalRel of R ) by A6, A7;
hence  X . i = Y . i by A8; ::_thesis: verum
end;
hence  X = Y by PBOOLE:3; ::_thesis: verum
end;
end;

:: deftheorem Def5 defines pcs-InternalRels PCS_0:def_5_:_
 for I being set
for C being RelStr-yielding ManySortedSet of I
for b3 being ManySortedSet of I holds
( b3 = pcs-InternalRels C iff for i being set st i in I holds
ex P being RelStr st
( P = C . i & b3 . i = the InternalRel of P ) );

definition
let I be non empty set ;
let C be RelStr-yielding ManySortedSet of I;
redefine func pcs-InternalRels C means :Def6: :: PCS_0:def 6
 for i being Element of I holds it . i = the InternalRel of (C . i);
compatibility
 for b1 being ManySortedSet of I holds
( b1 = pcs-InternalRels C iff for i being Element of I holds b1 . i = the InternalRel of (C . i) )
proof
let X be ManySortedSet of I; ::_thesis: ( X = pcs-InternalRels C iff for i being Element of I holds X . i = the InternalRel of (C . i) )
thus  ( X = pcs-InternalRels C implies for i being Element of I holds X . i = the InternalRel of (C . i) ) ::_thesis: ( ( for i being Element of I holds X . i = the InternalRel of (C . i) ) implies X = pcs-InternalRels C )
proof
assume A1: X = pcs-InternalRels C ; ::_thesis: for i being Element of I holds X . i = the InternalRel of (C . i)
let i be Element of I; ::_thesis: X . i = the InternalRel of (C . i)
 ex P being RelStr st
( P = C . i & X . i = the InternalRel of P ) by A1, Def5;
hence  X . i = the InternalRel of (C . i) ; ::_thesis: verum
end;
assume A2: for i being Element of I holds X . i = the InternalRel of (C . i) ; ::_thesis: X = pcs-InternalRels C
 for i being set st i in I holds
ex P being RelStr st
( P = C . i & X . i = the InternalRel of P )
proof
let i be set ; ::_thesis: ( i in I implies ex P being RelStr st
( P = C . i & X . i = the InternalRel of P ) )
assume  i in I ; ::_thesis: ex P being RelStr st
( P = C . i & X . i = the InternalRel of P )
then reconsider i = i as Element of I ;
take  C . i ; ::_thesis: ( C . i = C . i & X . i = the InternalRel of (C . i) )
thus  ( C . i = C . i & X . i = the InternalRel of (C . i) ) by A2; ::_thesis: verum
end;
hence  X = pcs-InternalRels C by Def5; ::_thesis: verum
end;
end;

:: deftheorem Def6 defines pcs-InternalRels PCS_0:def_6_:_
 for I being non empty set
for C being RelStr-yielding ManySortedSet of I
for b3 being ManySortedSet of I holds
( b3 = pcs-InternalRels C iff for i being Element of I holds b3 . i = the InternalRel of (C . i) );

registration
let I be set ;
let C be RelStr-yielding ManySortedSet of I;
cluster pcs-InternalRels C ->  Relation-yielding ;
coherence
 pcs-InternalRels C is Relation-yielding
proof
set IR = pcs-InternalRels C;
let i be set ; :: according to FUNCOP_1:def_11 ::_thesis: ( not i in proj1 (pcs-InternalRels C) or (pcs-InternalRels C) . i is set )
assume  i in dom (pcs-InternalRels C) ; ::_thesis: (pcs-InternalRels C) . i is set
then  ex P being RelStr st
( P = C . i & (pcs-InternalRels C) . i = the InternalRel of P ) by Def5;
hence  (pcs-InternalRels C) . i is set ; ::_thesis: verum
end;
end;

registration
let I be non empty set ;
let C be RelStr-yielding transitive-yielding ManySortedSet of I;
let i be Element of I;
clusterC . i ->  transitive for RelStr ;
coherence
 for b1 being RelStr st b1 = C . i holds
b1 is transitive
proof
 dom C = I by PARTFUN1:def_2;
then  C . i in rng C by FUNCT_1:3;
hence  for b1 being RelStr st b1 = C . i holds
b1 is transitive by Def4; ::_thesis: verum
end;
end;

begin

definition
attrc1 is strict ;
struct TolStr  ->  1-sorted ;
aggrTolStr(# carrier, ToleranceRel #) ->  TolStr ;
sel ToleranceRel c1 ->  Relation of the carrier of c1;
end;

definition
let P be TolStr ;
let p, q be Element of P;
predp (--) q means :Def7: :: PCS_0:def 7
[p,q] in the ToleranceRel of P;
end;

:: deftheorem Def7 defines (--) PCS_0:def_7_:_
 for P being TolStr
for p, q being Element of P holds
( p (--) q iff [p,q] in the ToleranceRel of P );

definition
let P be TolStr ;
attrP is pcs-tol-total  means :Def8: :: PCS_0:def 8
 the ToleranceRel of P is total ;
attrP is pcs-tol-reflexive  means :Def9: :: PCS_0:def 9
 the ToleranceRel of P is_reflexive_in the carrier of P;
attrP is pcs-tol-irreflexive  means :Def10: :: PCS_0:def 10
 the ToleranceRel of P is_irreflexive_in the carrier of P;
attrP is pcs-tol-symmetric  means :Def11: :: PCS_0:def 11
 the ToleranceRel of P is_symmetric_in the carrier of P;
end;

:: deftheorem Def8 defines pcs-tol-total PCS_0:def_8_:_
 for P being TolStr holds
( P is pcs-tol-total iff the ToleranceRel of P is total );

:: deftheorem Def9 defines pcs-tol-reflexive PCS_0:def_9_:_
 for P being TolStr holds
( P is pcs-tol-reflexive iff the ToleranceRel of P is_reflexive_in the carrier of P );

:: deftheorem Def10 defines pcs-tol-irreflexive PCS_0:def_10_:_
 for P being TolStr holds
( P is pcs-tol-irreflexive iff the ToleranceRel of P is_irreflexive_in the carrier of P );

:: deftheorem Def11 defines pcs-tol-symmetric PCS_0:def_11_:_
 for P being TolStr holds
( P is pcs-tol-symmetric iff the ToleranceRel of P is_symmetric_in the carrier of P );

definition
func emptyTolStr  ->  TolStr  equals :: PCS_0:def 12
 TolStr(# {},({} ({},{})) #);
coherence
 TolStr(# {},({} ({},{})) #) is TolStr ;
end;

:: deftheorem  defines emptyTolStr PCS_0:def_12_:_
 emptyTolStr = TolStr(# {},({} ({},{})) #);

registration
cluster emptyTolStr  ->  empty strict ;
coherence
 ( emptyTolStr is empty & emptyTolStr is strict ) ;
end;

theorem Th2: :: PCS_0:2
for P being TolStr st P is empty holds
TolStr(# the carrier of P, the ToleranceRel of P #) = emptyTolStr
proof
let P be TolStr ; ::_thesis: ( P is empty implies TolStr(# the carrier of P, the ToleranceRel of P #) = emptyTolStr )
assume  P is empty ; ::_thesis: TolStr(# the carrier of P, the ToleranceRel of P #) = emptyTolStr
then  the carrier of P = {} ;
hence  TolStr(# the carrier of P, the ToleranceRel of P #) = emptyTolStr ; ::_thesis: verum
end;

registration
cluster pcs-tol-reflexive  ->  pcs-tol-total for TolStr ;
coherence
 for b1 being TolStr st b1 is pcs-tol-reflexive holds
b1 is pcs-tol-total
proof
let P be TolStr ; ::_thesis: ( P is pcs-tol-reflexive implies P is pcs-tol-total )
assume  P is pcs-tol-reflexive ; ::_thesis: P is pcs-tol-total
then  the ToleranceRel of P is_reflexive_in the carrier of P by Def9;
then  dom the ToleranceRel of P = the carrier of P by ORDERS_1:13;
hence  the ToleranceRel of P is total by PARTFUN1:def_2; :: according to PCS_0:def_8 ::_thesis: verum
end;
end;

registration
cluster empty  ->  pcs-tol-reflexive pcs-tol-irreflexive pcs-tol-symmetric for TolStr ;
coherence
 for b1 being TolStr st b1 is empty holds
( b1 is pcs-tol-reflexive & b1 is pcs-tol-irreflexive & b1 is pcs-tol-symmetric )
proof
let P be TolStr ; ::_thesis: ( P is empty implies ( P is pcs-tol-reflexive & P is pcs-tol-irreflexive & P is pcs-tol-symmetric ) )
assume A1: P is empty ; ::_thesis: ( P is pcs-tol-reflexive & P is pcs-tol-irreflexive & P is pcs-tol-symmetric )
then  TolStr(# the carrier of P, the ToleranceRel of P #) = emptyTolStr by Th2;
then A2: the carrier of P = field the ToleranceRel of P ;
hence  the ToleranceRel of P is_reflexive_in the carrier of P by A1, RELAT_2:def_9; :: according to PCS_0:def_9 ::_thesis: ( P is pcs-tol-irreflexive & P is pcs-tol-symmetric )
thus  the ToleranceRel of P is_irreflexive_in the carrier of P by A1, A2, RELAT_2:def_10; :: according to PCS_0:def_10 ::_thesis: P is pcs-tol-symmetric
thus  the ToleranceRel of P is_symmetric_in the carrier of P by A1, A2, RELAT_2:def_11; :: according to PCS_0:def_11 ::_thesis: verum
end;
end;

registration
cluster empty strict for TolStr ;
existence
 ex b1 being TolStr st
( b1 is empty & b1 is strict )
proof
take  emptyTolStr ; ::_thesis: ( emptyTolStr is empty & emptyTolStr is strict )
thus  ( emptyTolStr is empty & emptyTolStr is strict ) ; ::_thesis: verum
end;
end;

registration
let P be pcs-tol-total TolStr ;
cluster the ToleranceRel of P ->  total ;
coherence
 the ToleranceRel of P is total by Def8;
end;

registration
let P be pcs-tol-reflexive TolStr ;
cluster the ToleranceRel of P ->  reflexive ;
coherence
 the ToleranceRel of P is reflexive
proof
set TR = the ToleranceRel of P;
A1: field the ToleranceRel of P = the carrier of P by ORDERS_1:12;
 the ToleranceRel of P is_reflexive_in the carrier of P by Def9;
hence  the ToleranceRel of P is reflexive by A1, RELAT_2:def_9; ::_thesis: verum
end;
end;

registration
let P be pcs-tol-irreflexive TolStr ;
cluster the ToleranceRel of P ->  irreflexive ;
coherence
 the ToleranceRel of P is irreflexive
proof
set TR = the ToleranceRel of P;
A1: the ToleranceRel of P is_irreflexive_in the carrier of P by Def10;
let x be set ; :: according to RELAT_2:def_2,RELAT_2:def_10 ::_thesis: ( not x in field the ToleranceRel of P or not [^,^] in the ToleranceRel of P )
assume  x in field the ToleranceRel of P ; ::_thesis: not [^,^] in the ToleranceRel of P
assume A2: [x,x] in the ToleranceRel of P ; ::_thesis: contradiction
then  x in dom the ToleranceRel of P by XTUPLE_0:def_12;
hence  contradiction by A1, A2, RELAT_2:def_2; ::_thesis: verum
end;
end;

registration
let P be pcs-tol-symmetric TolStr ;
cluster the ToleranceRel of P ->  symmetric ;
coherence
 the ToleranceRel of P is symmetric
proof
set TR = the ToleranceRel of P;
A1: the ToleranceRel of P is_symmetric_in the carrier of P by Def11;
let x, y be set ; :: according to RELAT_2:def_3,RELAT_2:def_11 ::_thesis: ( not x in field the ToleranceRel of P or not y in field the ToleranceRel of P or not [^,^] in the ToleranceRel of P or [^,^] in the ToleranceRel of P )
assume that
 x in field the ToleranceRel of P and
 y in field the ToleranceRel of P ; ::_thesis: ( not [^,^] in the ToleranceRel of P or [^,^] in the ToleranceRel of P )
assume A2: [x,y] in the ToleranceRel of P ; ::_thesis: [^,^] in the ToleranceRel of P
then A3: x in dom the ToleranceRel of P by XTUPLE_0:def_12;
 y in rng the ToleranceRel of P by A2, XTUPLE_0:def_13;
hence  [^,^] in the ToleranceRel of P by A1, A2, A3, RELAT_2:def_3; ::_thesis: verum
end;
end;

registration
let L be pcs-tol-total TolStr ;
cluster TolStr(# the carrier of L, the ToleranceRel of L #) ->  pcs-tol-total ;
coherence
 TolStr(# the carrier of L, the ToleranceRel of L #) is pcs-tol-total by Def8;
end;

definition
let P be pcs-tol-symmetric TolStr ;
let p, q be Element of P;
:: original: (--)
redefine predp (--) q;
symmetry
 for p, q being Element of P st (P,b1,b2) holds
(P,b2,b1)
proof
let x, y be Element of P; ::_thesis: ( (P,x,y) implies (P,y,x) )
assume A1: [x,y] in the ToleranceRel of P ; :: according to PCS_0:def_7 ::_thesis: (P,y,x)
then A2: x in the carrier of P by ZFMISC_1:87;
 the ToleranceRel of P is_symmetric_in the carrier of P by Def11;
hence  [y,x] in the ToleranceRel of P by A1, A2, RELAT_2:def_3; :: according to PCS_0:def_7 ::_thesis: verum
end;
end;

registration
let D be set ;
cluster TolStr(# D,(nabla D) #) ->  pcs-tol-reflexive pcs-tol-symmetric ;
coherence
 ( TolStr(# D,(nabla D) #) is pcs-tol-reflexive & TolStr(# D,(nabla D) #) is pcs-tol-symmetric )
proof
set P = TolStr(# D,(nabla D) #);
set TR = the ToleranceRel of TolStr(# D,(nabla D) #);
A1: field the ToleranceRel of TolStr(# D,(nabla D) #) = the carrier of TolStr(# D,(nabla D) #) by ORDERS_1:12;
hence  the ToleranceRel of TolStr(# D,(nabla D) #) is_reflexive_in the carrier of TolStr(# D,(nabla D) #) by RELAT_2:def_9; :: according to PCS_0:def_9 ::_thesis: TolStr(# D,(nabla D) #) is pcs-tol-symmetric
thus  the ToleranceRel of TolStr(# D,(nabla D) #) is_symmetric_in the carrier of TolStr(# D,(nabla D) #) by A1, RELAT_2:def_11; :: according to PCS_0:def_11 ::_thesis: verum
end;
end;

registration
let D be set ;
cluster TolStr(# D,({} (D,D)) #) ->  pcs-tol-irreflexive pcs-tol-symmetric ;
coherence
 ( TolStr(# D,({} (D,D)) #) is pcs-tol-irreflexive & TolStr(# D,({} (D,D)) #) is pcs-tol-symmetric )
proof
set P = TolStr(# D,({} (D,D)) #);
thus  the ToleranceRel of TolStr(# D,({} (D,D)) #) is_irreflexive_in the carrier of TolStr(# D,({} (D,D)) #) :: according to PCS_0:def_10 ::_thesis: TolStr(# D,({} (D,D)) #) is pcs-tol-symmetric
proof
let x be set ; :: according to RELAT_2:def_2 ::_thesis: ( not x in the carrier of TolStr(# D,({} (D,D)) #) or not [^,^] in the ToleranceRel of TolStr(# D,({} (D,D)) #) )
thus  ( not x in the carrier of TolStr(# D,({} (D,D)) #) or not [^,^] in the ToleranceRel of TolStr(# D,({} (D,D)) #) ) ; ::_thesis: verum
end;
let x be set ; :: according to RELAT_2:def_3,PCS_0:def_11 ::_thesis: for b1 being set holds
( not x in the carrier of TolStr(# D,({} (D,D)) #) or not b1 in the carrier of TolStr(# D,({} (D,D)) #) or not [^,^] in the ToleranceRel of TolStr(# D,({} (D,D)) #) or [^,^] in the ToleranceRel of TolStr(# D,({} (D,D)) #) )
thus  for b1 being set holds
( not x in the carrier of TolStr(# D,({} (D,D)) #) or not b1 in the carrier of TolStr(# D,({} (D,D)) #) or not [^,^] in the ToleranceRel of TolStr(# D,({} (D,D)) #) or [^,^] in the ToleranceRel of TolStr(# D,({} (D,D)) #) ) ; ::_thesis: verum
end;
end;

registration
cluster non empty strict pcs-tol-reflexive pcs-tol-symmetric for TolStr ;
existence
 ex b1 being TolStr st
( b1 is strict & not b1 is empty & b1 is pcs-tol-reflexive & b1 is pcs-tol-symmetric )
proof
take P = TolStr(# {{}},(nabla {{}}) #); ::_thesis: ( P is strict & not P is empty & P is pcs-tol-reflexive & P is pcs-tol-symmetric )
thus  P is strict ; ::_thesis: ( not P is empty & P is pcs-tol-reflexive & P is pcs-tol-symmetric )
thus  not the carrier of P is empty ; :: according to STRUCT_0:def_1 ::_thesis: ( P is pcs-tol-reflexive & P is pcs-tol-symmetric )
thus  ( P is pcs-tol-reflexive & P is pcs-tol-symmetric ) ; ::_thesis: verum
end;
end;

registration
cluster non empty strict pcs-tol-irreflexive pcs-tol-symmetric for TolStr ;
existence
 ex b1 being TolStr st
( b1 is strict & not b1 is empty & b1 is pcs-tol-irreflexive & b1 is pcs-tol-symmetric )
proof
take P = TolStr(# {{}},({} ({{}},{{}})) #); ::_thesis: ( P is strict & not P is empty & P is pcs-tol-irreflexive & P is pcs-tol-symmetric )
thus  P is strict ; ::_thesis: ( not P is empty & P is pcs-tol-irreflexive & P is pcs-tol-symmetric )
thus  not the carrier of P is empty ; :: according to STRUCT_0:def_1 ::_thesis: ( P is pcs-tol-irreflexive & P is pcs-tol-symmetric )
thus  ( P is pcs-tol-irreflexive & P is pcs-tol-symmetric ) ; ::_thesis: verum
end;
end;

definition
let R be Relation;
attrR is TolStr-yielding  means :Def13: :: PCS_0:def 13
 for P being set st P in rng R holds
P is TolStr ;
end;

:: deftheorem Def13 defines TolStr-yielding PCS_0:def_13_:_
 for R being Relation holds
( R is TolStr-yielding iff for P being set st P in rng R holds
P is TolStr );

definition
let f be Function;
redefine attr f is TolStr-yielding  means :Def14: :: PCS_0:def 14
 for x being set st x in dom f holds
f . x is TolStr ;
compatibility
 ( f is TolStr-yielding iff for x being set st x in dom f holds
f . x is TolStr )
proof
hereby  ::_thesis: ( ( for x being set st x in dom f holds
f . x is TolStr ) implies f is TolStr-yielding )
assume A1: f is TolStr-yielding ; ::_thesis: for x being set st x in dom f holds
f . x is TolStr
let x be set ; ::_thesis: ( x in dom f implies f . x is TolStr )
assume  x in dom f ; ::_thesis: f . x is TolStr
then  f . x in rng f by FUNCT_1:3;
hence  f . x is TolStr by A1, Def13; ::_thesis: verum
end;
assume A2: for x being set st x in dom f holds
f . x is TolStr ; ::_thesis: f is TolStr-yielding
let P be set ; :: according to PCS_0:def_13 ::_thesis: ( P in rng f implies P is TolStr )
assume  P in rng f ; ::_thesis: P is TolStr
then  ex x being set st
( x in dom f & f . x = P ) by FUNCT_1:def_3;
hence  P is TolStr by A2; ::_thesis: verum
end;
end;

:: deftheorem Def14 defines TolStr-yielding PCS_0:def_14_:_
 for f being Function holds
( f is TolStr-yielding iff for x being set st x in dom f holds
f . x is TolStr );

definition
let I be set ;
let f be ManySortedSet of I;
A1: dom f = I by PARTFUN1:def_2;
:: original: TolStr-yielding
redefine attrf is TolStr-yielding  means :: PCS_0:def 15
 for x being set st x in I holds
f . x is TolStr ;
compatibility
 ( f is TolStr-yielding iff for x being set st x in I holds
f . x is TolStr ) by A1, Def14;
end;

:: deftheorem  defines TolStr-yielding PCS_0:def_15_:_
 for I being set
for f being ManySortedSet of I holds
( f is TolStr-yielding iff for x being set st x in I holds
f . x is TolStr );

definition
let R be Relation;
attrR is pcs-tol-reflexive-yielding  means :Def16: :: PCS_0:def 16
 for S being TolStr st S in rng R holds
S is pcs-tol-reflexive ;
attrR is pcs-tol-irreflexive-yielding  means :Def17: :: PCS_0:def 17
 for S being TolStr st S in rng R holds
S is pcs-tol-irreflexive ;
attrR is pcs-tol-symmetric-yielding  means :Def18: :: PCS_0:def 18
 for S being TolStr st S in rng R holds
S is pcs-tol-symmetric ;
end;

:: deftheorem Def16 defines pcs-tol-reflexive-yielding PCS_0:def_16_:_
 for R being Relation holds
( R is pcs-tol-reflexive-yielding iff for S being TolStr st S in rng R holds
S is pcs-tol-reflexive );

:: deftheorem Def17 defines pcs-tol-irreflexive-yielding PCS_0:def_17_:_
 for R being Relation holds
( R is pcs-tol-irreflexive-yielding iff for S being TolStr st S in rng R holds
S is pcs-tol-irreflexive );

:: deftheorem Def18 defines pcs-tol-symmetric-yielding PCS_0:def_18_:_
 for R being Relation holds
( R is pcs-tol-symmetric-yielding iff for S being TolStr st S in rng R holds
S is pcs-tol-symmetric );

registration
cluster Relation-like empty  ->  pcs-tol-reflexive-yielding pcs-tol-irreflexive-yielding pcs-tol-symmetric-yielding for set ;
coherence
 for b1 being Relation st b1 is empty holds
( b1 is pcs-tol-reflexive-yielding & b1 is pcs-tol-irreflexive-yielding & b1 is pcs-tol-symmetric-yielding )
proof
let f be Relation; ::_thesis: ( f is empty implies ( f is pcs-tol-reflexive-yielding & f is pcs-tol-irreflexive-yielding & f is pcs-tol-symmetric-yielding ) )
assume A1: f is empty ; ::_thesis: ( f is pcs-tol-reflexive-yielding & f is pcs-tol-irreflexive-yielding & f is pcs-tol-symmetric-yielding )
thus  f is pcs-tol-reflexive-yielding ::_thesis: ( f is pcs-tol-irreflexive-yielding & f is pcs-tol-symmetric-yielding )
proof
let i be set ; :: according to PCS_0:def_16 ::_thesis: ( i is TolStr & i in rng f implies i is pcs-tol-reflexive )
thus  ( i is TolStr & i in rng f implies i is pcs-tol-reflexive ) by A1; ::_thesis: verum
end;
thus  f is pcs-tol-irreflexive-yielding ::_thesis: f is pcs-tol-symmetric-yielding
proof
let i be set ; :: according to PCS_0:def_17 ::_thesis: ( i is TolStr & i in rng f implies i is pcs-tol-irreflexive )
thus  ( i is TolStr & i in rng f implies i is pcs-tol-irreflexive ) by A1; ::_thesis: verum
end;
let i be set ; :: according to PCS_0:def_18 ::_thesis: ( i is TolStr & i in rng f implies i is pcs-tol-symmetric )
thus  ( i is TolStr & i in rng f implies i is pcs-tol-symmetric ) by A1; ::_thesis: verum
end;
end;

registration
let I be set ;
let P be TolStr ;
clusterI --> P -> () for ManySortedSet of I;
coherence
 for b1 being ManySortedSet of I st b1 = I --> P holds
b1 is TolStr-yielding
proof
 I --> P is ()
proof
let i be set ; :: according to PCS_0:def_15 ::_thesis: ( i in I implies (I --> P) . i is TolStr )
thus  ( i in I implies (I --> P) . i is TolStr ) by FUNCOP_1:7; ::_thesis: verum
end;
hence  for b1 being ManySortedSet of I st b1 = I --> P holds
b1 is TolStr-yielding ; ::_thesis: verum
end;
end;

registration
let I be set ;
let P be pcs-tol-reflexive TolStr ;
clusterI --> P ->  pcs-tol-reflexive-yielding for ManySortedSet of I;
coherence
 for b1 being ManySortedSet of I st b1 = I --> P holds
b1 is pcs-tol-reflexive-yielding
proof
set f = I --> P;
 I --> P is pcs-tol-reflexive-yielding
proof
let S be TolStr ; :: according to PCS_0:def_16 ::_thesis: ( S in rng (I --> P) implies S is pcs-tol-reflexive )
assume  S in rng (I --> P) ; ::_thesis: S is pcs-tol-reflexive
hence  S is pcs-tol-reflexive by TARSKI:def_1; ::_thesis: verum
end;
hence  for b1 being ManySortedSet of I st b1 = I --> P holds
b1 is pcs-tol-reflexive-yielding ; ::_thesis: verum
end;
end;

registration
let I be set ;
let P be pcs-tol-irreflexive TolStr ;
clusterI --> P ->  pcs-tol-irreflexive-yielding for ManySortedSet of I;
coherence
 for b1 being ManySortedSet of I st b1 = I --> P holds
b1 is pcs-tol-irreflexive-yielding
proof
set f = I --> P;
 I --> P is pcs-tol-irreflexive-yielding
proof
let S be TolStr ; :: according to PCS_0:def_17 ::_thesis: ( S in rng (I --> P) implies S is pcs-tol-irreflexive )
assume  S in rng (I --> P) ; ::_thesis: S is pcs-tol-irreflexive
hence  S is pcs-tol-irreflexive by TARSKI:def_1; ::_thesis: verum
end;
hence  for b1 being ManySortedSet of I st b1 = I --> P holds
b1 is pcs-tol-irreflexive-yielding ; ::_thesis: verum
end;
end;

registration
let I be set ;
let P be pcs-tol-symmetric TolStr ;
clusterI --> P ->  pcs-tol-symmetric-yielding for ManySortedSet of I;
coherence
 for b1 being ManySortedSet of I st b1 = I --> P holds
b1 is pcs-tol-symmetric-yielding
proof
set f = I --> P;
 I --> P is pcs-tol-symmetric-yielding
proof
let S be TolStr ; :: according to PCS_0:def_18 ::_thesis: ( S in rng (I --> P) implies S is pcs-tol-symmetric )
assume  S in rng (I --> P) ; ::_thesis: S is pcs-tol-symmetric
hence  S is pcs-tol-symmetric by TARSKI:def_1; ::_thesis: verum
end;
hence  for b1 being ManySortedSet of I st b1 = I --> P holds
b1 is pcs-tol-symmetric-yielding ; ::_thesis: verum
end;
end;

registration
cluster Relation-like Function-like TolStr-yielding  ->  1-sorted-yielding for set ;
coherence
 for b1 being Function st b1 is TolStr-yielding holds
b1 is 1-sorted-yielding
proof
let f be Function; ::_thesis: ( f is TolStr-yielding implies f is 1-sorted-yielding )
assume A1: f is TolStr-yielding ; ::_thesis: f is 1-sorted-yielding
let x be set ; :: according to PRALG_1:def_11 ::_thesis: ( not x in proj1 f or f . x is 1-sorted )
thus  ( not x in proj1 f or f . x is 1-sorted ) by A1, Def14; ::_thesis: verum
end;
end;

registration
let I be set ;
cluster Relation-like I -defined Function-like total () pcs-tol-reflexive-yielding pcs-tol-symmetric-yielding for set ;
existence
 ex b1 being ManySortedSet of I st
( b1 is pcs-tol-reflexive-yielding & b1 is pcs-tol-symmetric-yielding & b1 is () )
proof
take  I --> TolStr(# 0,(nabla 0) #) ; ::_thesis: ( I --> TolStr(# 0,(nabla 0) #) is pcs-tol-reflexive-yielding & I --> TolStr(# 0,(nabla 0) #) is pcs-tol-symmetric-yielding & I --> TolStr(# 0,(nabla 0) #) is () )
thus  ( I --> TolStr(# 0,(nabla 0) #) is pcs-tol-reflexive-yielding & I --> TolStr(# 0,(nabla 0) #) is pcs-tol-symmetric-yielding & I --> TolStr(# 0,(nabla 0) #) is () ) ; ::_thesis: verum
end;
end;

registration
let I be set ;
cluster Relation-like I -defined Function-like total () pcs-tol-irreflexive-yielding pcs-tol-symmetric-yielding for set ;
existence
 ex b1 being ManySortedSet of I st
( b1 is pcs-tol-irreflexive-yielding & b1 is pcs-tol-symmetric-yielding & b1 is () )
proof
take  I --> TolStr(# 0,({} (0,0)) #) ; ::_thesis: ( I --> TolStr(# 0,({} (0,0)) #) is pcs-tol-irreflexive-yielding & I --> TolStr(# 0,({} (0,0)) #) is pcs-tol-symmetric-yielding & I --> TolStr(# 0,({} (0,0)) #) is () )
thus  ( I --> TolStr(# 0,({} (0,0)) #) is pcs-tol-irreflexive-yielding & I --> TolStr(# 0,({} (0,0)) #) is pcs-tol-symmetric-yielding & I --> TolStr(# 0,({} (0,0)) #) is () ) ; ::_thesis: verum
end;
end;

registration
let I be set ;
cluster Relation-like I -defined Function-like total () for set ;
existence
 not for b1 being ManySortedSet of I holds b1 is ()
proof
set R = the TolStr ;
take  I --> the TolStr ; ::_thesis: I --> the TolStr is ()
thus  I --> the TolStr is () ; ::_thesis: verum
end;
end;

definition
let I be non empty set ;
let C be () ManySortedSet of I;
let i be Element of I;
:: original: .
redefine funcC . i ->  TolStr ;
coherence
 C . i is TolStr
proof
 dom C = I by PARTFUN1:def_2;
hence  C . i is TolStr by Def14; ::_thesis: verum
end;
end;

definition
let I be set ;
let C be () ManySortedSet of I;
func pcs-ToleranceRels C ->  ManySortedSet of I means :Def19: :: PCS_0:def 19
 for i being set st i in I holds
ex P being TolStr st
( P = C . i & it . i = the ToleranceRel of P );
existence
 ex b1 being ManySortedSet of I st
for i being set st i in I holds
ex P being TolStr st
( P = C . i & b1 . i = the ToleranceRel of P )
proof
defpred S1[ set , set ] means  ex R being TolStr st
( R = C . $1 & $2 = the ToleranceRel of R );
A1: for i being set st i in I holds
ex j being set st S1[i,j]
proof
let i be set ; ::_thesis: ( i in I implies ex j being set st S1[i,j] )
assume A2: i in I ; ::_thesis: ex j being set st S1[i,j]
then reconsider I = I as non empty set ;
reconsider B = C as () ManySortedSet of I ;
reconsider i9 = i as Element of I by A2;
take  the ToleranceRel of (B . i9) ; ::_thesis: S1[i, the ToleranceRel of (B . i9)]
take  B . i9 ; ::_thesis: ( B . i9 = C . i & the ToleranceRel of (B . i9) = the ToleranceRel of (B . i9) )
thus  ( B . i9 = C . i & the ToleranceRel of (B . i9) = the ToleranceRel of (B . i9) ) ; ::_thesis: verum
end;
consider M being Function such that
A3: dom M = I and
A4: for i being set st i in I holds
S1[i,M . i] from CLASSES1:sch_1(A1);
 M is ManySortedSet of I by A3, PARTFUN1:def_2, RELAT_1:def_18;
hence  ex b1 being ManySortedSet of I st
for i being set st i in I holds
ex P being TolStr st
( P = C . i & b1 . i = the ToleranceRel of P ) by A4; ::_thesis: verum
end;
uniqueness
 for b1, b2 being ManySortedSet of I st ( for i being set st i in I holds
ex P being TolStr st
( P = C . i & b1 . i = the ToleranceRel of P ) ) & ( for i being set st i in I holds
ex P being TolStr st
( P = C . i & b2 . i = the ToleranceRel of P ) ) holds
b1 = b2
proof
let X, Y be ManySortedSet of I; ::_thesis: ( ( for i being set st i in I holds
ex P being TolStr st
( P = C . i & X . i = the ToleranceRel of P ) ) & ( for i being set st i in I holds
ex P being TolStr st
( P = C . i & Y . i = the ToleranceRel of P ) ) implies X = Y )
assume that
A5: for j being set st j in I holds
ex R being TolStr st
( R = C . j & X . j = the ToleranceRel of R ) and
A6: for j being set st j in I holds
ex R being TolStr st
( R = C . j & Y . j = the ToleranceRel of R ) ; ::_thesis: X = Y
 for i being set st i in I holds
X . i = Y . i
proof
let i be set ; ::_thesis: ( i in I implies X . i = Y . i )
assume A7: i in I ; ::_thesis: X . i = Y . i
then A8: ex R being TolStr st
( R = C . i & X . i = the ToleranceRel of R ) by A5;
 ex R being TolStr st
( R = C . i & Y . i = the ToleranceRel of R ) by A6, A7;
hence  X . i = Y . i by A8; ::_thesis: verum
end;
hence  X = Y by PBOOLE:3; ::_thesis: verum
end;
end;

:: deftheorem Def19 defines pcs-ToleranceRels PCS_0:def_19_:_
 for I being set
for C being () ManySortedSet of I
for b3 being ManySortedSet of I holds
( b3 = pcs-ToleranceRels C iff for i being set st i in I holds
ex P being TolStr st
( P = C . i & b3 . i = the ToleranceRel of P ) );

definition
let I be non empty set ;
let C be () ManySortedSet of I;
redefine func pcs-ToleranceRels C means :Def20: :: PCS_0:def 20
 for i being Element of I holds it . i = the ToleranceRel of (C . i);
compatibility
 for b1 being ManySortedSet of I holds
( b1 = pcs-ToleranceRels C iff for i being Element of I holds b1 . i = the ToleranceRel of (C . i) )
proof
let X be ManySortedSet of I; ::_thesis: ( X = pcs-ToleranceRels C iff for i being Element of I holds X . i = the ToleranceRel of (C . i) )
thus  ( X = pcs-ToleranceRels C implies for i being Element of I holds X . i = the ToleranceRel of (C . i) ) ::_thesis: ( ( for i being Element of I holds X . i = the ToleranceRel of (C . i) ) implies X = pcs-ToleranceRels C )
proof
assume A1: X = pcs-ToleranceRels C ; ::_thesis: for i being Element of I holds X . i = the ToleranceRel of (C . i)
let i be Element of I; ::_thesis: X . i = the ToleranceRel of (C . i)
 ex P being TolStr st
( P = C . i & X . i = the ToleranceRel of P ) by A1, Def19;
hence  X . i = the ToleranceRel of (C . i) ; ::_thesis: verum
end;
assume A2: for i being Element of I holds X . i = the ToleranceRel of (C . i) ; ::_thesis: X = pcs-ToleranceRels C
 for i being set st i in I holds
ex P being TolStr st
( P = C . i & X . i = the ToleranceRel of P )
proof
let i be set ; ::_thesis: ( i in I implies ex P being TolStr st
( P = C . i & X . i = the ToleranceRel of P ) )
assume  i in I ; ::_thesis: ex P being TolStr st
( P = C . i & X . i = the ToleranceRel of P )
then reconsider i = i as Element of I ;
take  C . i ; ::_thesis: ( C . i = C . i & X . i = the ToleranceRel of (C . i) )
thus  ( C . i = C . i & X . i = the ToleranceRel of (C . i) ) by A2; ::_thesis: verum
end;
hence  X = pcs-ToleranceRels C by Def19; ::_thesis: verum
end;
end;

:: deftheorem Def20 defines pcs-ToleranceRels PCS_0:def_20_:_
 for I being non empty set
for C being () ManySortedSet of I
for b3 being ManySortedSet of I holds
( b3 = pcs-ToleranceRels C iff for i being Element of I holds b3 . i = the ToleranceRel of (C . i) );

registration
let I be set ;
let C be () ManySortedSet of I;
cluster pcs-ToleranceRels C ->  Relation-yielding ;
coherence
 pcs-ToleranceRels C is Relation-yielding
proof
set TR = pcs-ToleranceRels C;
let i be set ; :: according to FUNCOP_1:def_11 ::_thesis: ( not i in proj1 (pcs-ToleranceRels C) or (pcs-ToleranceRels C) . i is set )
assume  i in dom (pcs-ToleranceRels C) ; ::_thesis: (pcs-ToleranceRels C) . i is set
then  ex P being TolStr st
( P = C . i & (pcs-ToleranceRels C) . i = the ToleranceRel of P ) by Def19;
hence  (pcs-ToleranceRels C) . i is set ; ::_thesis: verum
end;
end;

registration
let I be non empty set ;
let C be () pcs-tol-reflexive-yielding ManySortedSet of I;
let i be Element of I;
clusterC . i ->  pcs-tol-reflexive for TolStr ;
coherence
 for b1 being TolStr st b1 = C . i holds
b1 is pcs-tol-reflexive
proof
 dom C = I by PARTFUN1:def_2;
then  C . i in rng C by FUNCT_1:3;
hence  for b1 being TolStr st b1 = C . i holds
b1 is pcs-tol-reflexive by Def16; ::_thesis: verum
end;
end;

registration
let I be non empty set ;
let C be () pcs-tol-irreflexive-yielding ManySortedSet of I;
let i be Element of I;
clusterC . i ->  pcs-tol-irreflexive for TolStr ;
coherence
 for b1 being TolStr st b1 = C . i holds
b1 is pcs-tol-irreflexive
proof
 dom C = I by PARTFUN1:def_2;
then  C . i in rng C by FUNCT_1:3;
hence  for b1 being TolStr st b1 = C . i holds
b1 is pcs-tol-irreflexive by Def17; ::_thesis: verum
end;
end;

registration
let I be non empty set ;
let C be () pcs-tol-symmetric-yielding ManySortedSet of I;
let i be Element of I;
clusterC . i ->  pcs-tol-symmetric for TolStr ;
coherence
 for b1 being TolStr st b1 = C . i holds
b1 is pcs-tol-symmetric
proof
 dom C = I by PARTFUN1:def_2;
then  C . i in rng C by FUNCT_1:3;
hence  for b1 being TolStr st b1 = C . i holds
b1 is pcs-tol-symmetric by Def18; ::_thesis: verum
end;
end;

theorem Th3: :: PCS_0:3
for P, Q being TolStr st TolStr(# the carrier of P, the ToleranceRel of P #) = TolStr(# the carrier of Q, the ToleranceRel of Q #) & P is pcs-tol-reflexive holds
Q is pcs-tol-reflexive
proof
let P, Q be TolStr ; ::_thesis: ( TolStr(# the carrier of P, the ToleranceRel of P #) = TolStr(# the carrier of Q, the ToleranceRel of Q #) & P is pcs-tol-reflexive implies Q is pcs-tol-reflexive )
assume that
A1: TolStr(# the carrier of P, the ToleranceRel of P #) = TolStr(# the carrier of Q, the ToleranceRel of Q #) and
A2: the ToleranceRel of P is_reflexive_in the carrier of P ; :: according to PCS_0:def_9 ::_thesis: Q is pcs-tol-reflexive
let x be set ; :: according to RELAT_2:def_1,PCS_0:def_9 ::_thesis: ( not x in the carrier of Q or [^,^] in the ToleranceRel of Q )
assume  x in the carrier of Q ; ::_thesis: [^,^] in the ToleranceRel of Q
hence  [^,^] in the ToleranceRel of Q by A1, A2, RELAT_2:def_1; ::_thesis: verum
end;

theorem Th4: :: PCS_0:4
for P, Q being TolStr st TolStr(# the carrier of P, the ToleranceRel of P #) = TolStr(# the carrier of Q, the ToleranceRel of Q #) & P is pcs-tol-irreflexive holds
Q is pcs-tol-irreflexive
proof
let P, Q be TolStr ; ::_thesis: ( TolStr(# the carrier of P, the ToleranceRel of P #) = TolStr(# the carrier of Q, the ToleranceRel of Q #) & P is pcs-tol-irreflexive implies Q is pcs-tol-irreflexive )
assume that
A1: TolStr(# the carrier of P, the ToleranceRel of P #) = TolStr(# the carrier of Q, the ToleranceRel of Q #) and
A2: the ToleranceRel of P is_irreflexive_in the carrier of P ; :: according to PCS_0:def_10 ::_thesis: Q is pcs-tol-irreflexive
let x be set ; :: according to RELAT_2:def_2,PCS_0:def_10 ::_thesis: ( not x in the carrier of Q or not [^,^] in the ToleranceRel of Q )
assume  x in the carrier of Q ; ::_thesis: not [^,^] in the ToleranceRel of Q
hence  not [^,^] in the ToleranceRel of Q by A1, A2, RELAT_2:def_2; ::_thesis: verum
end;

theorem Th5: :: PCS_0:5
for P, Q being TolStr st TolStr(# the carrier of P, the ToleranceRel of P #) = TolStr(# the carrier of Q, the ToleranceRel of Q #) & P is pcs-tol-symmetric holds
Q is pcs-tol-symmetric
proof
let P, Q be TolStr ; ::_thesis: ( TolStr(# the carrier of P, the ToleranceRel of P #) = TolStr(# the carrier of Q, the ToleranceRel of Q #) & P is pcs-tol-symmetric implies Q is pcs-tol-symmetric )
assume that
A1: TolStr(# the carrier of P, the ToleranceRel of P #) = TolStr(# the carrier of Q, the ToleranceRel of Q #) and
A2: the ToleranceRel of P is_symmetric_in the carrier of P ; :: according to PCS_0:def_11 ::_thesis: Q is pcs-tol-symmetric
let x, y be set ; :: according to RELAT_2:def_3,PCS_0:def_11 ::_thesis: ( not x in the carrier of Q or not y in the carrier of Q or not [^,^] in the ToleranceRel of Q or [^,^] in the ToleranceRel of Q )
assume  x in the carrier of Q ; ::_thesis: ( not y in the carrier of Q or not [^,^] in the ToleranceRel of Q or [^,^] in the ToleranceRel of Q )
hence  ( not y in the carrier of Q or not [^,^] in the ToleranceRel of Q or [^,^] in the ToleranceRel of Q ) by A1, A2, RELAT_2:def_3; ::_thesis: verum
end;

definition
let P, Q be TolStr ;
func[^P,Q^] ->  TolStr  equals :: PCS_0:def 21
 TolStr(# [: the carrier of P, the carrier of Q:],[^ the ToleranceRel of P, the ToleranceRel of Q^] #);
coherence
 TolStr(# [: the carrier of P, the carrier of Q:],[^ the ToleranceRel of P, the ToleranceRel of Q^] #) is TolStr ;
end;

:: deftheorem  defines [^ PCS_0:def_21_:_
 for P, Q being TolStr holds [^P,Q^] = TolStr(# [: the carrier of P, the carrier of Q:],[^ the ToleranceRel of P, the ToleranceRel of Q^] #);

notation
let P, Q be TolStr ;
let p be Element of P;
let q be Element of Q;
synonym [^p,q^] for [P,Q];
end;

definition
let P, Q be non empty TolStr ;
let p be Element of P;
let q be Element of Q;
:: original: [^
redefine func[^p,q^] ->  Element of [^P,Q^];
coherence
 [^,^] is Element of [^P,Q^]
proof
 [p,q] is Element of [^P,Q^] ;
hence  [^,^] is Element of [^P,Q^] ; ::_thesis: verum
end;
end;

notation
let P, Q be TolStr ;
let p be Element of [^P,Q^];
synonym p ^`1 for P `1 ;
synonym p ^`2 for P `2 ;
end;

definition
let P, Q be non empty TolStr ;
let p be Element of [^P,Q^];
:: original: ^`1
redefine funcp ^`1  ->  Element of P;
coherence
 ^`1 is Element of P by MCART_1:10;
:: original: ^`2
redefine funcp ^`2  ->  Element of Q;
coherence
 ^`2 is Element of Q by MCART_1:10;
end;

theorem Th6: :: PCS_0:6
for S1, S2 being non empty TolStr
for a, c being Element of S1
for b, d being Element of S2 holds
( [^a,b^] (--) [^c,d^] iff ( a (--) c or b (--) d ) )
proof
let S1, S2 be non empty TolStr ; ::_thesis: for a, c being Element of S1
for b, d being Element of S2 holds
( [^a,b^] (--) [^c,d^] iff ( a (--) c or b (--) d ) )
let a, c be Element of S1; ::_thesis: for b, d being Element of S2 holds
( [^a,b^] (--) [^c,d^] iff ( a (--) c or b (--) d ) )
let b, d be Element of S2; ::_thesis: ( [^a,b^] (--) [^c,d^] iff ( a (--) c or b (--) d ) )
set I1 = the ToleranceRel of S1;
set I2 = the ToleranceRel of S2;
set x = [[a,b],[c,d]];
thus  ( not [^a,b^] (--) [^c,d^] or a (--) c or b (--) d ) ::_thesis: ( ( a (--) c or b (--) d ) implies [^a,b^] (--) [^c,d^] )
proof
assume  [^a,b^] (--) [^c,d^] ; ::_thesis: ( a (--) c or b (--) d )
then  [[a,b],[c,d]] in the ToleranceRel of [^S1,S2^] by Def7;
then  ( [a,c] in the ToleranceRel of S1 or [b,d] in the ToleranceRel of S2 ) by Def3;
hence  ( a (--) c or b (--) d ) by Def7; ::_thesis: verum
end;
assume  ( a (--) c or b (--) d ) ; ::_thesis: [^a,b^] (--) [^c,d^]
then  ( [(([[a,b],[c,d]] `1) `1),(([[a,b],[c,d]] `2) `1)] in the ToleranceRel of S1 or [(([[a,b],[c,d]] `1) `2),(([[a,b],[c,d]] `2) `2)] in the ToleranceRel of S2 ) by Def7;
hence  [[a,b],[c,d]] in the ToleranceRel of [^S1,S2^] by Def3; :: according to PCS_0:def_7 ::_thesis: verum
end;

theorem :: PCS_0:7
for S1, S2 being non empty TolStr
for x, y being Element of [^S1,S2^] holds
( x (--) y iff ( x ^`1 (--) y ^`1 or x ^`2 (--) y ^`2 ) )
proof
let S1, S2 be non empty TolStr ; ::_thesis: for x, y being Element of [^S1,S2^] holds
( x (--) y iff ( x ^`1 (--) y ^`1 or x ^`2 (--) y ^`2 ) )
let x, y be Element of [^S1,S2^]; ::_thesis: ( x (--) y iff ( x ^`1 (--) y ^`1 or x ^`2 (--) y ^`2 ) )
A1: ex a, b being set st
( a in the carrier of S1 & b in the carrier of S2 & x = [a,b] ) by ZFMISC_1:def_2;
A2: ex c, d being set st
( c in the carrier of S1 & d in the carrier of S2 & y = [c,d] ) by ZFMISC_1:def_2;
A3: x = [(x ^`1),(x ^`2)] by A1, MCART_1:8;
 y = [(y ^`1),(y ^`2)] by A2, MCART_1:8;
hence  ( x (--) y iff ( x ^`1 (--) y ^`1 or x ^`2 (--) y ^`2 ) ) by A3, Th6; ::_thesis: verum
end;

registration
let P be TolStr ;
let Q be pcs-tol-reflexive TolStr ;
cluster[^P,Q^] ->  pcs-tol-reflexive ;
coherence
 [^P,Q^] is pcs-tol-reflexive
proof
let x be set ; :: according to RELAT_2:def_1,PCS_0:def_9 ::_thesis: ( not x in the carrier of [^P,Q^] or [^,^] in the ToleranceRel of [^P,Q^] )
assume  x in the carrier of [^P,Q^] ; ::_thesis: [^,^] in the ToleranceRel of [^P,Q^]
then consider x1, x2 being set  such that
A1: x1 in the carrier of P and
A2: x2 in the carrier of Q and
A3: x = [x1,x2] by ZFMISC_1:def_2;
reconsider D2 = the carrier of Q as non empty set by A2;
reconsider TQ = the ToleranceRel of Q as Relation of D2 ;
 D2 = field TQ by ORDERS_1:12;
then  TQ is_reflexive_in D2 by RELAT_2:def_9;
then  [x2,x2] in TQ by A2, RELAT_2:def_1;
hence  [^,^] in the ToleranceRel of [^P,Q^] by A1, A2, A3, Def3; ::_thesis: verum
end;
end;

registration
let P be pcs-tol-reflexive TolStr ;
let Q be TolStr ;
cluster[^P,Q^] ->  pcs-tol-reflexive ;
coherence
 [^P,Q^] is pcs-tol-reflexive
proof
let x be set ; :: according to RELAT_2:def_1,PCS_0:def_9 ::_thesis: ( not x in the carrier of [^P,Q^] or [^,^] in the ToleranceRel of [^P,Q^] )
assume  x in the carrier of [^P,Q^] ; ::_thesis: [^,^] in the ToleranceRel of [^P,Q^]
then consider x1, x2 being set  such that
A1: x1 in the carrier of P and
A2: x2 in the carrier of Q and
A3: x = [x1,x2] by ZFMISC_1:def_2;
reconsider D1 = the carrier of P as non empty set by A1;
reconsider TP = the ToleranceRel of P as Relation of D1 ;
 D1 = field TP by ORDERS_1:12;
then  TP is_reflexive_in D1 by RELAT_2:def_9;
then  [x1,x1] in TP by A1, RELAT_2:def_1;
hence  [^,^] in the ToleranceRel of [^P,Q^] by A1, A2, A3, Def3; ::_thesis: verum
end;
end;

registration
let P, Q be pcs-tol-symmetric TolStr ;
cluster[^P,Q^] ->  pcs-tol-symmetric ;
coherence
 [^P,Q^] is pcs-tol-symmetric
proof
set R = [^P,Q^];
set TR = the ToleranceRel of [^P,Q^];
A1: the ToleranceRel of [^P,Q^] is_symmetric_in field the ToleranceRel of [^P,Q^] by RELAT_2:def_11;
let x, y be set ; :: according to RELAT_2:def_3,PCS_0:def_11 ::_thesis: ( not x in the carrier of [^P,Q^] or not y in the carrier of [^P,Q^] or not [^,^] in the ToleranceRel of [^P,Q^] or [^,^] in the ToleranceRel of [^P,Q^] )
assume that
 x in the carrier of [^P,Q^] and
 y in the carrier of [^P,Q^] ; ::_thesis: ( not [^,^] in the ToleranceRel of [^P,Q^] or [^,^] in the ToleranceRel of [^P,Q^] )
assume A2: [x,y] in the ToleranceRel of [^P,Q^] ; ::_thesis: [^,^] in the ToleranceRel of [^P,Q^]
then A3: x in field the ToleranceRel of [^P,Q^] by RELAT_1:15;
 y in field the ToleranceRel of [^P,Q^] by A2, RELAT_1:15;
hence  [^,^] in the ToleranceRel of [^P,Q^] by A1, A2, A3, RELAT_2:def_3; ::_thesis: verum
end;
end;

begin

definition
attrc1 is strict ;
struct pcs-Str  ->  RelStr , TolStr ;
aggrpcs-Str(# carrier, InternalRel, ToleranceRel #) ->  pcs-Str ;
end;

definition
let P be pcs-Str ;
attrP is pcs-compatible  means :Def22: :: PCS_0:def 22
 for p, p9, q, q9 being Element of P st p (--) q & p9 <= p & q9 <= q holds
p9 (--) q9;
end;

:: deftheorem Def22 defines pcs-compatible PCS_0:def_22_:_
 for P being pcs-Str holds
( P is pcs-compatible iff for p, p9, q, q9 being Element of P st p (--) q & p9 <= p & q9 <= q holds
p9 (--) q9 );

definition
let P be pcs-Str ;
attrP is pcs-like  means :Def23: :: PCS_0:def 23
( P is reflexive & P is transitive & P is pcs-tol-reflexive & P is pcs-tol-symmetric & P is pcs-compatible );
attrP is anti-pcs-like  means :Def24: :: PCS_0:def 24
( P is reflexive & P is transitive & P is pcs-tol-irreflexive & P is pcs-tol-symmetric & P is pcs-compatible );
end;

:: deftheorem Def23 defines pcs-like PCS_0:def_23_:_
 for P being pcs-Str holds
( P is pcs-like iff ( P is reflexive & P is transitive & P is pcs-tol-reflexive & P is pcs-tol-symmetric & P is pcs-compatible ) );

:: deftheorem Def24 defines anti-pcs-like PCS_0:def_24_:_
 for P being pcs-Str holds
( P is anti-pcs-like iff ( P is reflexive & P is transitive & P is pcs-tol-irreflexive & P is pcs-tol-symmetric & P is pcs-compatible ) );

registration
cluster pcs-like  ->  reflexive transitive pcs-tol-reflexive pcs-tol-symmetric pcs-compatible for pcs-Str ;
coherence
 for b1 being pcs-Str st b1 is pcs-like holds
( b1 is reflexive & b1 is transitive & b1 is pcs-tol-reflexive & b1 is pcs-tol-symmetric & b1 is pcs-compatible ) by Def23;
cluster reflexive transitive pcs-tol-reflexive pcs-tol-symmetric pcs-compatible  ->  pcs-like for pcs-Str ;
coherence
 for b1 being pcs-Str st b1 is reflexive & b1 is transitive & b1 is pcs-tol-reflexive & b1 is pcs-tol-symmetric & b1 is pcs-compatible holds
b1 is pcs-like by Def23;
cluster anti-pcs-like  ->  reflexive transitive pcs-tol-irreflexive pcs-tol-symmetric pcs-compatible for pcs-Str ;
coherence
 for b1 being pcs-Str st b1 is anti-pcs-like holds
( b1 is reflexive & b1 is transitive & b1 is pcs-tol-irreflexive & b1 is pcs-tol-symmetric & b1 is pcs-compatible ) by Def24;
cluster reflexive transitive pcs-tol-irreflexive pcs-tol-symmetric pcs-compatible  ->  anti-pcs-like for pcs-Str ;
coherence
 for b1 being pcs-Str st b1 is reflexive & b1 is transitive & b1 is pcs-tol-irreflexive & b1 is pcs-tol-symmetric & b1 is pcs-compatible holds
b1 is anti-pcs-like by Def24;
end;

definition
let D be set ;
func pcs-total D ->  pcs-Str  equals :: PCS_0:def 25
 pcs-Str(# D,(nabla D),(nabla D) #);
coherence
 pcs-Str(# D,(nabla D),(nabla D) #) is pcs-Str ;
end;

:: deftheorem  defines pcs-total PCS_0:def_25_:_
 for D being set holds pcs-total D = pcs-Str(# D,(nabla D),(nabla D) #);

registration
let D be set ;
cluster pcs-total D ->  strict ;
coherence
 pcs-total D is strict ;
end;

registration
let D be non empty set ;
cluster pcs-total D ->  non empty ;
coherence
 not pcs-total D is empty ;
end;

registration
let D be set ;
cluster pcs-total D ->  reflexive transitive pcs-tol-reflexive pcs-tol-symmetric ;
coherence
 ( pcs-total D is reflexive & pcs-total D is transitive & pcs-total D is pcs-tol-reflexive & pcs-total D is pcs-tol-symmetric )
proof
set P = pcs-total D;
set IR = the InternalRel of (pcs-total D);
set TR = the ToleranceRel of (pcs-total D);
A1: field the InternalRel of (pcs-total D) = the carrier of (pcs-total D) by ORDERS_1:12;
hence  the InternalRel of (pcs-total D) is_reflexive_in the carrier of (pcs-total D) by RELAT_2:def_9; :: according to ORDERS_2:def_2 ::_thesis: ( pcs-total D is transitive & pcs-total D is pcs-tol-reflexive & pcs-total D is pcs-tol-symmetric )
thus  the InternalRel of (pcs-total D) is_transitive_in the carrier of (pcs-total D) by A1, RELAT_2:def_16; :: according to ORDERS_2:def_3 ::_thesis: ( pcs-total D is pcs-tol-reflexive & pcs-total D is pcs-tol-symmetric )
thus  the ToleranceRel of (pcs-total D) is_reflexive_in the carrier of (pcs-total D) by A1, RELAT_2:def_9; :: according to PCS_0:def_9 ::_thesis: pcs-total D is pcs-tol-symmetric
thus  the ToleranceRel of (pcs-total D) is_symmetric_in the carrier of (pcs-total D) by A1, RELAT_2:def_11; :: according to PCS_0:def_11 ::_thesis: verum
end;
end;

registration
let D be set ;
cluster pcs-total D ->  pcs-like ;
coherence
 pcs-total D is pcs-like
proof
set P = pcs-total D;
thus  ( pcs-total D is reflexive & pcs-total D is transitive ) ; :: according to PCS_0:def_23 ::_thesis: ( pcs-total D is pcs-tol-reflexive & pcs-total D is pcs-tol-symmetric & pcs-total D is pcs-compatible )
thus  ( pcs-total D is pcs-tol-reflexive & pcs-total D is pcs-tol-symmetric ) ; ::_thesis: pcs-total D is pcs-compatible
let p, p9, q, q9 be Element of (pcs-total D); :: according to PCS_0:def_22 ::_thesis: ( p (--) q & p9 <= p & q9 <= q implies p9 (--) q9 )
assume  p (--) q ; ::_thesis: ( not p9 <= p or not q9 <= q or p9 (--) q9 )
assume that
A1: p9 <= p and
 q9 <= q ; ::_thesis: p9 (--) q9
 [p9,p] in [:D,D:] by A1, ORDERS_2:def_5;
then  p9 in the carrier of (pcs-total D) by ZFMISC_1:87;
hence  [p9,q9] in the ToleranceRel of (pcs-total D) by ZFMISC_1:87; :: according to PCS_0:def_7 ::_thesis: verum
end;
end;

registration
let D be set ;
cluster pcs-Str(# D,(nabla D),({} (D,D)) #) ->  anti-pcs-like ;
coherence
 pcs-Str(# D,(nabla D),({} (D,D)) #) is anti-pcs-like
proof
set P = pcs-Str(# D,(nabla D),({} (D,D)) #);
A1: RelStr(# the carrier of pcs-Str(# D,(nabla D),({} (D,D)) #), the InternalRel of pcs-Str(# D,(nabla D),({} (D,D)) #) #) = RelStr(# the carrier of RelStr(# D,(nabla D) #), the InternalRel of RelStr(# D,(nabla D) #) #) ;
hence  pcs-Str(# D,(nabla D),({} (D,D)) #) is reflexive by WAYBEL_8:12; :: according to PCS_0:def_24 ::_thesis: ( pcs-Str(# D,(nabla D),({} (D,D)) #) is transitive & pcs-Str(# D,(nabla D),({} (D,D)) #) is pcs-tol-irreflexive & pcs-Str(# D,(nabla D),({} (D,D)) #) is pcs-tol-symmetric & pcs-Str(# D,(nabla D),({} (D,D)) #) is pcs-compatible )
thus  pcs-Str(# D,(nabla D),({} (D,D)) #) is transitive by A1, WAYBEL_8:13; ::_thesis: ( pcs-Str(# D,(nabla D),({} (D,D)) #) is pcs-tol-irreflexive & pcs-Str(# D,(nabla D),({} (D,D)) #) is pcs-tol-symmetric & pcs-Str(# D,(nabla D),({} (D,D)) #) is pcs-compatible )
A2: TolStr(# the carrier of pcs-Str(# D,(nabla D),({} (D,D)) #), the ToleranceRel of pcs-Str(# D,(nabla D),({} (D,D)) #) #) = TolStr(# the carrier of TolStr(# D,({} (D,D)) #), the ToleranceRel of TolStr(# D,({} (D,D)) #) #) ;
hence  pcs-Str(# D,(nabla D),({} (D,D)) #) is pcs-tol-irreflexive by Th4; ::_thesis: ( pcs-Str(# D,(nabla D),({} (D,D)) #) is pcs-tol-symmetric & pcs-Str(# D,(nabla D),({} (D,D)) #) is pcs-compatible )
thus  pcs-Str(# D,(nabla D),({} (D,D)) #) is pcs-tol-symmetric by A2, Th5; ::_thesis: pcs-Str(# D,(nabla D),({} (D,D)) #) is pcs-compatible
let p be Element of pcs-Str(# D,(nabla D),({} (D,D)) #); :: according to PCS_0:def_22 ::_thesis: for p9, q, q9 being Element of pcs-Str(# D,(nabla D),({} (D,D)) #) st p (--) q & p9 <= p & q9 <= q holds
p9 (--) q9
thus  for p9, q, q9 being Element of pcs-Str(# D,(nabla D),({} (D,D)) #) st p (--) q & p9 <= p & q9 <= q holds
p9 (--) q9 by Def7; ::_thesis: verum
end;
end;

registration
cluster non empty strict pcs-like for pcs-Str ;
existence
 ex b1 being pcs-Str st
( b1 is strict & not b1 is empty & b1 is pcs-like )
proof
take P = pcs-total {{}}; ::_thesis: ( P is strict & not P is empty & P is pcs-like )
thus  P is strict ; ::_thesis: ( not P is empty & P is pcs-like )
thus  not the carrier of P is empty ; :: according to STRUCT_0:def_1 ::_thesis: P is pcs-like
thus  P is pcs-like ; ::_thesis: verum
end;
cluster non empty strict anti-pcs-like for pcs-Str ;
existence
 ex b1 being pcs-Str st
( b1 is strict & not b1 is empty & b1 is anti-pcs-like )
proof
take P = pcs-Str(# {{}},(nabla {{}}),({} ({{}},{{}})) #); ::_thesis: ( P is strict & not P is empty & P is anti-pcs-like )
thus  P is strict ; ::_thesis: ( not P is empty & P is anti-pcs-like )
thus  not the carrier of P is empty ; :: according to STRUCT_0:def_1 ::_thesis: P is anti-pcs-like
thus  P is anti-pcs-like ; ::_thesis: verum
end;
end;

definition
mode pcs is pcs-like pcs-Str ;
mode anti-pcs is anti-pcs-like pcs-Str ;
end;

definition
func pcs-empty  ->  pcs-Str  equals :: PCS_0:def 26
 pcs-total 0;
coherence
 pcs-total 0 is pcs-Str ;
end;

:: deftheorem  defines pcs-empty PCS_0:def_26_:_
 pcs-empty = pcs-total 0;

registration
cluster pcs-empty  ->  empty strict pcs-like ;
coherence
 ( pcs-empty is strict & pcs-empty is empty & pcs-empty is pcs-like ) ;
end;

definition
let p be set ;
func pcs-singleton p ->  pcs-Str  equals :: PCS_0:def 27
 pcs-total {p};
coherence
 pcs-total {p} is pcs-Str ;
end;

:: deftheorem  defines pcs-singleton PCS_0:def_27_:_
 for p being set holds pcs-singleton p = pcs-total {p};

registration
let p be set ;
cluster pcs-singleton p ->  non empty strict pcs-like ;
coherence
 ( pcs-singleton p is strict & not pcs-singleton p is empty & pcs-singleton p is pcs-like ) ;
end;

definition
let R be Relation;
attrR is pcs-Str-yielding  means :Def28: :: PCS_0:def 28
 for P being set st P in rng R holds
P is pcs-Str ;
attrR is pcs-yielding  means :Def29: :: PCS_0:def 29
 for P being set st P in rng R holds
P is pcs;
end;

:: deftheorem Def28 defines pcs-Str-yielding PCS_0:def_28_:_
 for R being Relation holds
( R is pcs-Str-yielding iff for P being set st P in rng R holds
P is pcs-Str );

:: deftheorem Def29 defines pcs-yielding PCS_0:def_29_:_
 for R being Relation holds
( R is pcs-yielding iff for P being set st P in rng R holds
P is pcs );

definition
let f be Function;
redefine attr f is pcs-Str-yielding  means :Def30: :: PCS_0:def 30
 for x being set st x in dom f holds
f . x is pcs-Str ;
compatibility
 ( f is pcs-Str-yielding iff for x being set st x in dom f holds
f . x is pcs-Str )
proof
hereby  ::_thesis: ( ( for x being set st x in dom f holds
f . x is pcs-Str ) implies f is pcs-Str-yielding )
assume A1: f is pcs-Str-yielding ; ::_thesis: for x being set st x in dom f holds
f . x is pcs-Str
let x be set ; ::_thesis: ( x in dom f implies f . x is pcs-Str )
assume  x in dom f ; ::_thesis: f . x is pcs-Str
then  f . x in rng f by FUNCT_1:3;
hence  f . x is pcs-Str by A1, Def28; ::_thesis: verum
end;
assume A2: for x being set st x in dom f holds
f . x is pcs-Str ; ::_thesis: f is pcs-Str-yielding
let P be set ; :: according to PCS_0:def_28 ::_thesis: ( P in rng f implies P is pcs-Str )
assume  P in rng f ; ::_thesis: P is pcs-Str
then  ex x being set st
( x in dom f & f . x = P ) by FUNCT_1:def_3;
hence  P is pcs-Str by A2; ::_thesis: verum
end;
redefine attr f is pcs-yielding  means :Def31: :: PCS_0:def 31
 for x being set st x in dom f holds
f . x is pcs;
compatibility
 ( f is pcs-yielding iff for x being set st x in dom f holds
f . x is pcs )
proof
hereby  ::_thesis: ( ( for x being set st x in dom f holds
f . x is pcs ) implies f is pcs-yielding )
assume A3: f is pcs-yielding ; ::_thesis: for x being set st x in dom f holds
f . x is pcs
let x be set ; ::_thesis: ( x in dom f implies f . x is pcs )
assume  x in dom f ; ::_thesis: f . x is pcs
then  f . x in rng f by FUNCT_1:3;
hence  f . x is pcs by A3, Def29; ::_thesis: verum
end;
assume A4: for x being set st x in dom f holds
f . x is pcs ; ::_thesis: f is pcs-yielding
let P be set ; :: according to PCS_0:def_29 ::_thesis: ( P in rng f implies P is pcs )
assume  P in rng f ; ::_thesis: P is pcs
then  ex x being set st
( x in dom f & f . x = P ) by FUNCT_1:def_3;
hence  P is pcs by A4; ::_thesis: verum
end;
end;

:: deftheorem Def30 defines pcs-Str-yielding PCS_0:def_30_:_
 for f being Function holds
( f is pcs-Str-yielding iff for x being set st x in dom f holds
f . x is pcs-Str );

:: deftheorem Def31 defines pcs-yielding PCS_0:def_31_:_
 for f being Function holds
( f is pcs-yielding iff for x being set st x in dom f holds
f . x is pcs );

definition
let I be set ;
let f be ManySortedSet of I;
A1: dom f = I by PARTFUN1:def_2;
:: original: pcs-Str-yielding
redefine attrf is pcs-Str-yielding  means :Def32: :: PCS_0:def 32
 for x being set st x in I holds
f . x is pcs-Str ;
compatibility
 ( f is pcs-Str-yielding iff for x being set st x in I holds
f . x is pcs-Str ) by A1, Def30;
:: original: pcs-yielding
redefine attrf is pcs-yielding  means :Def33: :: PCS_0:def 33
 for x being set st x in I holds
f . x is pcs;
compatibility
 ( f is pcs-yielding iff for x being set st x in I holds
f . x is pcs ) by A1, Def31;
end;

:: deftheorem Def32 defines pcs-Str-yielding PCS_0:def_32_:_
 for I being set
for f being ManySortedSet of I holds
( f is pcs-Str-yielding iff for x being set st x in I holds
f . x is pcs-Str );

:: deftheorem Def33 defines pcs-yielding PCS_0:def_33_:_
 for I being set
for f being ManySortedSet of I holds
( f is pcs-yielding iff for x being set st x in I holds
f . x is pcs );

registration
cluster Relation-like pcs-Str-yielding  ->  RelStr-yielding TolStr-yielding for set ;
coherence
 for b1 being Relation st b1 is pcs-Str-yielding holds
( b1 is TolStr-yielding & b1 is RelStr-yielding )
proof
let f be Relation; ::_thesis: ( f is pcs-Str-yielding implies ( f is TolStr-yielding & f is RelStr-yielding ) )
assume A1: f is pcs-Str-yielding ; ::_thesis: ( f is TolStr-yielding & f is RelStr-yielding )
thus  f is TolStr-yielding ::_thesis: f is RelStr-yielding
proof
let y be set ; :: according to PCS_0:def_13 ::_thesis: ( y in rng f implies y is TolStr )
thus  ( y in rng f implies y is TolStr ) by A1, Def28; ::_thesis: verum
end;
let y be set ; :: according to YELLOW_1:def_3 ::_thesis: ( not y in proj2 f or y is RelStr )
thus  ( not y in proj2 f or y is RelStr ) by A1, Def28; ::_thesis: verum
end;
cluster Relation-like pcs-yielding  ->  pcs-Str-yielding for set ;
coherence
 for b1 being Relation st b1 is pcs-yielding holds
b1 is pcs-Str-yielding
proof
let f be Relation; ::_thesis: ( f is pcs-yielding implies f is pcs-Str-yielding )
assume A2: f is pcs-yielding ; ::_thesis: f is pcs-Str-yielding
let y be set ; :: according to PCS_0:def_28 ::_thesis: ( y in rng f implies y is pcs-Str )
thus  ( y in rng f implies y is pcs-Str ) by A2, Def29; ::_thesis: verum
end;
cluster Relation-like pcs-yielding  ->  reflexive-yielding transitive-yielding pcs-tol-reflexive-yielding pcs-tol-symmetric-yielding for set ;
coherence
 for b1 being Relation st b1 is pcs-yielding holds
( b1 is reflexive-yielding & b1 is transitive-yielding & b1 is pcs-tol-reflexive-yielding & b1 is pcs-tol-symmetric-yielding )
proof
let f be Relation; ::_thesis: ( f is pcs-yielding implies ( f is reflexive-yielding & f is transitive-yielding & f is pcs-tol-reflexive-yielding & f is pcs-tol-symmetric-yielding ) )
assume A3: f is pcs-yielding ; ::_thesis: ( f is reflexive-yielding & f is transitive-yielding & f is pcs-tol-reflexive-yielding & f is pcs-tol-symmetric-yielding )
thus  f is reflexive-yielding ::_thesis: ( f is transitive-yielding & f is pcs-tol-reflexive-yielding & f is pcs-tol-symmetric-yielding )
proof
let y be RelStr ; :: according to WAYBEL_3:def_8 ::_thesis: ( not y in proj2 f or y is reflexive )
thus  ( not y in proj2 f or y is reflexive ) by A3, Def29; ::_thesis: verum
end;
thus  f is transitive-yielding ::_thesis: ( f is pcs-tol-reflexive-yielding & f is pcs-tol-symmetric-yielding )
proof
let y be RelStr ; :: according to PCS_0:def_4 ::_thesis: ( y in rng f implies y is transitive )
thus  ( y in rng f implies y is transitive ) by A3, Def29; ::_thesis: verum
end;
thus  f is pcs-tol-reflexive-yielding ::_thesis: f is pcs-tol-symmetric-yielding
proof
let y be TolStr ; :: according to PCS_0:def_16 ::_thesis: ( y in rng f implies y is pcs-tol-reflexive )
thus  ( y in rng f implies y is pcs-tol-reflexive ) by A3, Def29; ::_thesis: verum
end;
let y be TolStr ; :: according to PCS_0:def_18 ::_thesis: ( y in rng f implies y is pcs-tol-symmetric )
thus  ( y in rng f implies y is pcs-tol-symmetric ) by A3, Def29; ::_thesis: verum
end;
end;

registration
let I be set ;
let P be pcs;
clusterI --> P -> () for ManySortedSet of I;
coherence
 for b1 being ManySortedSet of I st b1 = I --> P holds
b1 is pcs-yielding
proof
 I --> P is ()
proof
let i be set ; :: according to PCS_0:def_33 ::_thesis: ( i in I implies (I --> P) . i is pcs )
thus  ( i in I implies (I --> P) . i is pcs ) by FUNCOP_1:7; ::_thesis: verum
end;
hence  for b1 being ManySortedSet of I st b1 = I --> P holds
b1 is pcs-yielding ; ::_thesis: verum
end;
end;

registration
let I be set ;
cluster Relation-like I -defined Function-like total () for set ;
existence
 not for b1 being ManySortedSet of I holds b1 is ()
proof
take  I --> pcs-empty ; ::_thesis: I --> pcs-empty is ()
thus  I --> pcs-empty is () ; ::_thesis: verum
end;
end;

definition
let I be non empty set ;
let C be () ManySortedSet of I;
let i be Element of I;
:: original: .
redefine funcC . i ->  pcs-Str ;
coherence
 C . i is pcs-Str by Def32;
end;

definition
let I be non empty set ;
let C be () ManySortedSet of I;
let i be Element of I;
:: original: .
redefine funcC . i ->  pcs;
coherence
 C . i is pcs by Def33;
end;

definition
let P, Q be pcs-Str ;
predP c= Q means :Def34: :: PCS_0:def 34
( the carrier of P c= the carrier of Q & the InternalRel of P c= the InternalRel of Q & the ToleranceRel of P c= the ToleranceRel of Q );
reflexivity
 for P being pcs-Str holds
( the carrier of P c= the carrier of P & the InternalRel of P c= the InternalRel of P & the ToleranceRel of P c= the ToleranceRel of P ) ;
end;

:: deftheorem Def34 defines c= PCS_0:def_34_:_
 for P, Q being pcs-Str holds
( P c= Q iff ( the carrier of P c= the carrier of Q & the InternalRel of P c= the InternalRel of Q & the ToleranceRel of P c= the ToleranceRel of Q ) );

theorem Th8: :: PCS_0:8
for P, Q being RelStr
for p, q being Element of P
for p1, q1 being Element of Q st the InternalRel of P c= the InternalRel of Q & p = p1 & q = q1 & p <= q holds
p1 <= q1
proof
let P, Q be RelStr ; ::_thesis: for p, q being Element of P
for p1, q1 being Element of Q st the InternalRel of P c= the InternalRel of Q & p = p1 & q = q1 & p <= q holds
p1 <= q1
let p, q be Element of P; ::_thesis: for p1, q1 being Element of Q st the InternalRel of P c= the InternalRel of Q & p = p1 & q = q1 & p <= q holds
p1 <= q1
let p1, q1 be Element of Q; ::_thesis: ( the InternalRel of P c= the InternalRel of Q & p = p1 & q = q1 & p <= q implies p1 <= q1 )
assume that
A1: the InternalRel of P c= the InternalRel of Q and
A2: p = p1 and
A3: q = q1 and
A4: [p,q] in the InternalRel of P ; :: according to ORDERS_2:def_5 ::_thesis: p1 <= q1
thus  [p1,q1] in the InternalRel of Q by A1, A2, A3, A4; :: according to ORDERS_2:def_5 ::_thesis: verum
end;

theorem Th9: :: PCS_0:9
for P, Q being TolStr
for p, q being Element of P
for p1, q1 being Element of Q st the ToleranceRel of P c= the ToleranceRel of Q & p = p1 & q = q1 & p (--) q holds
p1 (--) q1
proof
let P, Q be TolStr ; ::_thesis: for p, q being Element of P
for p1, q1 being Element of Q st the ToleranceRel of P c= the ToleranceRel of Q & p = p1 & q = q1 & p (--) q holds
p1 (--) q1
let p, q be Element of P; ::_thesis: for p1, q1 being Element of Q st the ToleranceRel of P c= the ToleranceRel of Q & p = p1 & q = q1 & p (--) q holds
p1 (--) q1
let p1, q1 be Element of Q; ::_thesis: ( the ToleranceRel of P c= the ToleranceRel of Q & p = p1 & q = q1 & p (--) q implies p1 (--) q1 )
assume that
A1: the ToleranceRel of P c= the ToleranceRel of Q and
A2: p = p1 and
A3: q = q1 and
A4: [p,q] in the ToleranceRel of P ; :: according to PCS_0:def_7 ::_thesis: p1 (--) q1
thus  [p1,q1] in the ToleranceRel of Q by A1, A2, A3, A4; :: according to PCS_0:def_7 ::_thesis: verum
end;

Lm2:  for P, Q being pcs-Str
for p being set st p in the carrier of P & P c= Q holds
p is Element of Q
proof
let P, Q be pcs-Str ; ::_thesis: for p being set st p in the carrier of P & P c= Q holds
p is Element of Q
let p be set ; ::_thesis: ( p in the carrier of P & P c= Q implies p is Element of Q )
assume A1: p in the carrier of P ; ::_thesis: ( not P c= Q or p is Element of Q )
assume  P c= Q ; ::_thesis: p is Element of Q
then  the carrier of P c= the carrier of Q by Def34;
hence  p is Element of Q by A1; ::_thesis: verum
end;

definition
let C be Relation;
attrC is pcs-chain-like  means :Def35: :: PCS_0:def 35
 for P, Q being pcs-Str st P in rng C & Q in rng C & not P c= Q holds
Q c= P;
end;

:: deftheorem Def35 defines pcs-chain-like PCS_0:def_35_:_
 for C being Relation holds
( C is pcs-chain-like iff for P, Q being pcs-Str st P in rng C & Q in rng C & not P c= Q holds
Q c= P );

registration
let I be set ;
let P be pcs-Str ;
clusterI --> P ->  pcs-chain-like for ManySortedSet of I;
coherence
 for b1 being ManySortedSet of I st b1 = I --> P holds
b1 is pcs-chain-like
proof
set f = I --> P;
 I --> P is pcs-chain-like
proof
let R, S be pcs-Str ; :: according to PCS_0:def_35 ::_thesis: ( R in rng (I --> P) & S in rng (I --> P) & not R c= S implies S c= R )
assume that
A1: R in rng (I --> P) and
A2: S in rng (I --> P) ; ::_thesis: ( R c= S or S c= R )
 ( ( P = R & P = S ) or rng (I --> P) = {} ) by A1, A2, TARSKI:def_1;
hence  ( R c= S or S c= R ) by A1; ::_thesis: verum
end;
hence  for b1 being ManySortedSet of I st b1 = I --> P holds
b1 is pcs-chain-like ; ::_thesis: verum
end;
end;

registration
cluster Relation-like Function-like pcs-yielding pcs-chain-like for set ;
existence
 ex b1 being Function st
( b1 is pcs-chain-like & b1 is pcs-yielding )
proof
set P = the pcs;
take  0 --> the pcs ; ::_thesis: ( 0 --> the pcs is pcs-chain-like & 0 --> the pcs is pcs-yielding )
thus  ( 0 --> the pcs is pcs-chain-like & 0 --> the pcs is pcs-yielding ) ; ::_thesis: verum
end;
end;

registration
let I be set ;
cluster Relation-like I -defined Function-like total () pcs-chain-like for set ;
existence
 ex b1 being ManySortedSet of I st
( b1 is pcs-chain-like & b1 is () )
proof
set P = the pcs;
take  I --> the pcs ; ::_thesis: ( I --> the pcs is pcs-chain-like & I --> the pcs is () )
thus  ( I --> the pcs is pcs-chain-like & I --> the pcs is () ) ; ::_thesis: verum
end;
end;

definition
let I be set ;
mode pcs-Chain of I is () pcs-chain-like ManySortedSet of I;
end;

definition
let I be set ;
let C be () ManySortedSet of I;
func pcs-union C ->  strict pcs-Str  means :Def36: :: PCS_0:def 36
( the carrier of it = Union (Carrier C) & the InternalRel of it = Union (pcs-InternalRels C) & the ToleranceRel of it = Union (pcs-ToleranceRels C) );
existence
 ex b1 being strict pcs-Str st
( the carrier of b1 = Union (Carrier C) & the InternalRel of b1 = Union (pcs-InternalRels C) & the ToleranceRel of b1 = Union (pcs-ToleranceRels C) )
proof
set CA = Carrier C;
set IRA = pcs-InternalRels C;
set TRA = pcs-ToleranceRels C;
set D = Union (Carrier C);
set IR = Union (pcs-InternalRels C);
set TR = Union (pcs-ToleranceRels C);
A1: dom (Carrier C) = I by PARTFUN1:def_2;
 Union (pcs-InternalRels C) c= [:(Union (Carrier C)),(Union (Carrier C)):]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Union (pcs-InternalRels C) or x in [:(Union (Carrier C)),(Union (Carrier C)):] )
assume  x in Union (pcs-InternalRels C) ; ::_thesis: x in [:(Union (Carrier C)),(Union (Carrier C)):]
then consider P being set  such that
A2: x in P and
A3: P in rng (pcs-InternalRels C) by TARSKI:def_4;
consider i being set  such that
A4: i in dom (pcs-InternalRels C) and
A5: (pcs-InternalRels C) . i = P by A3, FUNCT_1:def_3;
consider R being RelStr  such that
A6: R = C . i and
A7: (pcs-InternalRels C) . i = the InternalRel of R by A4, Def5;
consider x1, x2 being set  such that
A8: x = [x1,x2] and
A9: x1 in the carrier of R and
A10: x2 in the carrier of R by A2, A5, A7, RELSET_1:2;
 ex S being 1-sorted st
( S = C . i & (Carrier C) . i = the carrier of S ) by A4, PRALG_1:def_13;
then A11: the carrier of R in rng (Carrier C) by A1, A4, A6, FUNCT_1:def_3;
then A12: x1 in union (rng (Carrier C)) by A9, TARSKI:def_4;
 x2 in union (rng (Carrier C)) by A10, A11, TARSKI:def_4;
hence  x in [:(Union (Carrier C)),(Union (Carrier C)):] by A8, A12, ZFMISC_1:87; ::_thesis: verum
end;
then reconsider IR = Union (pcs-InternalRels C) as Relation of (Union (Carrier C)) ;
 Union (pcs-ToleranceRels C) c= [:(Union (Carrier C)),(Union (Carrier C)):]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Union (pcs-ToleranceRels C) or x in [:(Union (Carrier C)),(Union (Carrier C)):] )
assume  x in Union (pcs-ToleranceRels C) ; ::_thesis: x in [:(Union (Carrier C)),(Union (Carrier C)):]
then consider P being set  such that
A13: x in P and
A14: P in rng (pcs-ToleranceRels C) by TARSKI:def_4;
consider i being set  such that
A15: i in dom (pcs-ToleranceRels C) and
A16: (pcs-ToleranceRels C) . i = P by A14, FUNCT_1:def_3;
consider R being TolStr  such that
A17: R = C . i and
A18: (pcs-ToleranceRels C) . i = the ToleranceRel of R by A15, Def19;
consider x1, x2 being set  such that
A19: x = [x1,x2] and
A20: x1 in the carrier of R and
A21: x2 in the carrier of R by A13, A16, A18, RELSET_1:2;
 ex S being 1-sorted st
( S = C . i & (Carrier C) . i = the carrier of S ) by A15, PRALG_1:def_13;
then A22: the carrier of R in rng (Carrier C) by A1, A15, A17, FUNCT_1:def_3;
then A23: x1 in union (rng (Carrier C)) by A20, TARSKI:def_4;
 x2 in union (rng (Carrier C)) by A21, A22, TARSKI:def_4;
hence  x in [:(Union (Carrier C)),(Union (Carrier C)):] by A19, A23, ZFMISC_1:87; ::_thesis: verum
end;
then reconsider TR = Union (pcs-ToleranceRels C) as Relation of (Union (Carrier C)) ;
take  pcs-Str(# (Union (Carrier C)),IR,TR #) ; ::_thesis: ( the carrier of pcs-Str(# (Union (Carrier C)),IR,TR #) = Union (Carrier C) & the InternalRel of pcs-Str(# (Union (Carrier C)),IR,TR #) = Union (pcs-InternalRels C) & the ToleranceRel of pcs-Str(# (Union (Carrier C)),IR,TR #) = Union (pcs-ToleranceRels C) )
thus  ( the carrier of pcs-Str(# (Union (Carrier C)),IR,TR #) = Union (Carrier C) & the InternalRel of pcs-Str(# (Union (Carrier C)),IR,TR #) = Union (pcs-InternalRels C) & the ToleranceRel of pcs-Str(# (Union (Carrier C)),IR,TR #) = Union (pcs-ToleranceRels C) ) ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being strict pcs-Str st the carrier of b1 = Union (Carrier C) & the InternalRel of b1 = Union (pcs-InternalRels C) & the ToleranceRel of b1 = Union (pcs-ToleranceRels C) & the carrier of b2 = Union (Carrier C) & the InternalRel of b2 = Union (pcs-InternalRels C) & the ToleranceRel of b2 = Union (pcs-ToleranceRels C) holds
b1 = b2 ;
end;

:: deftheorem Def36 defines pcs-union PCS_0:def_36_:_
 for I being set
for C being () ManySortedSet of I
for b3 being strict pcs-Str holds
( b3 = pcs-union C iff ( the carrier of b3 = Union (Carrier C) & the InternalRel of b3 = Union (pcs-InternalRels C) & the ToleranceRel of b3 = Union (pcs-ToleranceRels C) ) );

theorem Th10: :: PCS_0:10
for I being set
for C being () ManySortedSet of I
for p, q being Element of (pcs-union C) holds
( p <= q iff ex i being set ex P being pcs-Str ex p9, q9 being Element of P st
( i in I & P = C . i & p9 = p & q9 = q & p9 <= q9 ) )
proof
let I be set ; ::_thesis: for C being () ManySortedSet of I
for p, q being Element of (pcs-union C) holds
( p <= q iff ex i being set ex P being pcs-Str ex p9, q9 being Element of P st
( i in I & P = C . i & p9 = p & q9 = q & p9 <= q9 ) )
let C be () ManySortedSet of I; ::_thesis: for p, q being Element of (pcs-union C) holds
( p <= q iff ex i being set ex P being pcs-Str ex p9, q9 being Element of P st
( i in I & P = C . i & p9 = p & q9 = q & p9 <= q9 ) )
set R = pcs-union C;
let p, q be Element of (pcs-union C); ::_thesis: ( p <= q iff ex i being set ex P being pcs-Str ex p9, q9 being Element of P st
( i in I & P = C . i & p9 = p & q9 = q & p9 <= q9 ) )
A1: dom (pcs-InternalRels C) = I by PARTFUN1:def_2;
thus  ( p <= q implies ex i being set ex P being pcs-Str ex p9, q9 being Element of P st
( i in I & P = C . i & p9 = p & q9 = q & p9 <= q9 ) ) ::_thesis: ( ex i being set ex P being pcs-Str ex p9, q9 being Element of P st
( i in I & P = C . i & p9 = p & q9 = q & p9 <= q9 ) implies p <= q )
proof
assume  p <= q ; ::_thesis: ex i being set ex P being pcs-Str ex p9, q9 being Element of P st
( i in I & P = C . i & p9 = p & q9 = q & p9 <= q9 )
then  [p,q] in the InternalRel of (pcs-union C) by ORDERS_2:def_5;
then  [p,q] in Union (pcs-InternalRels C) by Def36;
then consider Z being set  such that
A2: [p,q] in Z and
A3: Z in rng (pcs-InternalRels C) by TARSKI:def_4;
consider i being set  such that
A4: i in dom (pcs-InternalRels C) and
A5: (pcs-InternalRels C) . i = Z by A3, FUNCT_1:def_3;
reconsider I1 = I as non empty set by A4;
reconsider A1 = C as () ManySortedSet of I1 ;
reconsider i1 = i as Element of I1 by A4;
reconsider P = A1 . i1 as pcs-Str ;
take  i ; ::_thesis: ex P being pcs-Str ex p9, q9 being Element of P st
( i in I & P = C . i & p9 = p & q9 = q & p9 <= q9 )
take  P ; ::_thesis: ex p9, q9 being Element of P st
( i in I & P = C . i & p9 = p & q9 = q & p9 <= q9 )
 Z = the InternalRel of (A1 . i1) by A5, Def6;
then reconsider p9 = p, q9 = q as Element of P by A2, ZFMISC_1:87;
take  p9 ; ::_thesis: ex q9 being Element of P st
( i in I & P = C . i & p9 = p & q9 = q & p9 <= q9 )
take  q9 ; ::_thesis: ( i in I & P = C . i & p9 = p & q9 = q & p9 <= q9 )
thus  i in I by A4; ::_thesis: ( P = C . i & p9 = p & q9 = q & p9 <= q9 )
thus  ( P = C . i & p9 = p & q9 = q ) ; ::_thesis: p9 <= q9
thus  [p9,q9] in the InternalRel of P by A2, A5, Def6; :: according to ORDERS_2:def_5 ::_thesis: verum
end;
given i being set , P being pcs-Str , p9, q9 being Element of P such that A6: i in I and
A7: P = C . i and
A8: p9 = p and
A9: q9 = q and
A10: p9 <= q9 ; ::_thesis: p <= q
A11: [p9,q9] in the InternalRel of P by A10, ORDERS_2:def_5;
reconsider I1 = I as non empty set by A6;
reconsider i1 = i as Element of I1 by A6;
reconsider A1 = C as () ManySortedSet of I1 ;
 (pcs-InternalRels A1) . i1 = the InternalRel of (A1 . i1) by Def6;
then  the InternalRel of (A1 . i1) in rng (pcs-InternalRels C) by A1, FUNCT_1:3;
then  [p,q] in Union (pcs-InternalRels C) by A7, A8, A9, A11, TARSKI:def_4;
hence  [p,q] in the InternalRel of (pcs-union C) by Def36; :: according to ORDERS_2:def_5 ::_thesis: verum
end;

theorem :: PCS_0:11
for I being non empty set
for C being () ManySortedSet of I
for p, q being Element of (pcs-union C) holds
( p <= q iff ex i being Element of I ex p9, q9 being Element of (C . i) st
( p9 = p & q9 = q & p9 <= q9 ) )
proof
let I be non empty set ; ::_thesis: for C being () ManySortedSet of I
for p, q being Element of (pcs-union C) holds
( p <= q iff ex i being Element of I ex p9, q9 being Element of (C . i) st
( p9 = p & q9 = q & p9 <= q9 ) )
let C be () ManySortedSet of I; ::_thesis: for p, q being Element of (pcs-union C) holds
( p <= q iff ex i being Element of I ex p9, q9 being Element of (C . i) st
( p9 = p & q9 = q & p9 <= q9 ) )
let p, q be Element of (pcs-union C); ::_thesis: ( p <= q iff ex i being Element of I ex p9, q9 being Element of (C . i) st
( p9 = p & q9 = q & p9 <= q9 ) )
thus  ( p <= q implies ex i being Element of I ex p9, q9 being Element of (C . i) st
( p9 = p & q9 = q & p9 <= q9 ) ) ::_thesis: ( ex i being Element of I ex p9, q9 being Element of (C . i) st
( p9 = p & q9 = q & p9 <= q9 ) implies p <= q )
proof
assume  p <= q ; ::_thesis: ex i being Element of I ex p9, q9 being Element of (C . i) st
( p9 = p & q9 = q & p9 <= q9 )
then  ex i being set ex P being pcs-Str ex p9, q9 being Element of P st
( i in I & P = C . i & p9 = p & q9 = q & p9 <= q9 ) by Th10;
hence  ex i being Element of I ex p9, q9 being Element of (C . i) st
( p9 = p & q9 = q & p9 <= q9 ) ; ::_thesis: verum
end;
thus  ( ex i being Element of I ex p9, q9 being Element of (C . i) st
( p9 = p & q9 = q & p9 <= q9 ) implies p <= q ) by Th10; ::_thesis: verum
end;

theorem Th12: :: PCS_0:12
for I being set
for C being () ManySortedSet of I
for p, q being Element of (pcs-union C) holds
( p (--) q iff ex i being set ex P being pcs-Str ex p9, q9 being Element of P st
( i in I & P = C . i & p9 = p & q9 = q & p9 (--) q9 ) )
proof
let I be set ; ::_thesis: for C being () ManySortedSet of I
for p, q being Element of (pcs-union C) holds
( p (--) q iff ex i being set ex P being pcs-Str ex p9, q9 being Element of P st
( i in I & P = C . i & p9 = p & q9 = q & p9 (--) q9 ) )
let C be () ManySortedSet of I; ::_thesis: for p, q being Element of (pcs-union C) holds
( p (--) q iff ex i being set ex P being pcs-Str ex p9, q9 being Element of P st
( i in I & P = C . i & p9 = p & q9 = q & p9 (--) q9 ) )
set R = pcs-union C;
let p, q be Element of (pcs-union C); ::_thesis: ( p (--) q iff ex i being set ex P being pcs-Str ex p9, q9 being Element of P st
( i in I & P = C . i & p9 = p & q9 = q & p9 (--) q9 ) )
A1: dom (pcs-ToleranceRels C) = I by PARTFUN1:def_2;
thus  ( p (--) q implies ex i being set ex P being pcs-Str ex p9, q9 being Element of P st
( i in I & P = C . i & p9 = p & q9 = q & p9 (--) q9 ) ) ::_thesis: ( ex i being set ex P being pcs-Str ex p9, q9 being Element of P st
( i in I & P = C . i & p9 = p & q9 = q & p9 (--) q9 ) implies p (--) q )
proof
assume  p (--) q ; ::_thesis: ex i being set ex P being pcs-Str ex p9, q9 being Element of P st
( i in I & P = C . i & p9 = p & q9 = q & p9 (--) q9 )
then  [p,q] in the ToleranceRel of (pcs-union C) by Def7;
then  [p,q] in Union (pcs-ToleranceRels C) by Def36;
then consider Z being set  such that
A2: [p,q] in Z and
A3: Z in rng (pcs-ToleranceRels C) by TARSKI:def_4;
consider i being set  such that
A4: i in dom (pcs-ToleranceRels C) and
A5: (pcs-ToleranceRels C) . i = Z by A3, FUNCT_1:def_3;
reconsider I1 = I as non empty set by A4;
reconsider A1 = C as () ManySortedSet of I1 ;
reconsider i1 = i as Element of I1 by A4;
reconsider P = A1 . i1 as pcs-Str ;
take  i ; ::_thesis: ex P being pcs-Str ex p9, q9 being Element of P st
( i in I & P = C . i & p9 = p & q9 = q & p9 (--) q9 )
take  P ; ::_thesis: ex p9, q9 being Element of P st
( i in I & P = C . i & p9 = p & q9 = q & p9 (--) q9 )
 Z = the ToleranceRel of (A1 . i1) by A5, Def20;
then reconsider p9 = p, q9 = q as Element of P by A2, ZFMISC_1:87;
take  p9 ; ::_thesis: ex q9 being Element of P st
( i in I & P = C . i & p9 = p & q9 = q & p9 (--) q9 )
take  q9 ; ::_thesis: ( i in I & P = C . i & p9 = p & q9 = q & p9 (--) q9 )
thus  i in I by A4; ::_thesis: ( P = C . i & p9 = p & q9 = q & p9 (--) q9 )
thus  ( P = C . i & p9 = p & q9 = q ) ; ::_thesis: p9 (--) q9
thus  [p9,q9] in the ToleranceRel of P by A2, A5, Def20; :: according to PCS_0:def_7 ::_thesis: verum
end;
given i being set , P being pcs-Str , p9, q9 being Element of P such that A6: i in I and
A7: P = C . i and
A8: p9 = p and
A9: q9 = q and
A10: p9 (--) q9 ; ::_thesis: p (--) q
A11: [p9,q9] in the ToleranceRel of P by A10, Def7;
reconsider I1 = I as non empty set by A6;
reconsider i1 = i as Element of I1 by A6;
reconsider A1 = C as () ManySortedSet of I1 ;
 (pcs-ToleranceRels A1) . i1 = the ToleranceRel of (A1 . i1) by Def20;
then  the ToleranceRel of (A1 . i1) in rng (pcs-ToleranceRels C) by A1, FUNCT_1:3;
then  [p,q] in Union (pcs-ToleranceRels C) by A7, A8, A9, A11, TARSKI:def_4;
hence  [p,q] in the ToleranceRel of (pcs-union C) by Def36; :: according to PCS_0:def_7 ::_thesis: verum
end;

theorem :: PCS_0:13
for I being non empty set
for C being () ManySortedSet of I
for p, q being Element of (pcs-union C) holds
( p (--) q iff ex i being Element of I ex p9, q9 being Element of (C . i) st
( p9 = p & q9 = q & p9 (--) q9 ) )
proof
let I be non empty set ; ::_thesis: for C being () ManySortedSet of I
for p, q being Element of (pcs-union C) holds
( p (--) q iff ex i being Element of I ex p9, q9 being Element of (C . i) st
( p9 = p & q9 = q & p9 (--) q9 ) )
let C be () ManySortedSet of I; ::_thesis: for p, q being Element of (pcs-union C) holds
( p (--) q iff ex i being Element of I ex p9, q9 being Element of (C . i) st
( p9 = p & q9 = q & p9 (--) q9 ) )
let p, q be Element of (pcs-union C); ::_thesis: ( p (--) q iff ex i being Element of I ex p9, q9 being Element of (C . i) st
( p9 = p & q9 = q & p9 (--) q9 ) )
thus  ( p (--) q implies ex i being Element of I ex p9, q9 being Element of (C . i) st
( p9 = p & q9 = q & p9 (--) q9 ) ) ::_thesis: ( ex i being Element of I ex p9, q9 being Element of (C . i) st
( p9 = p & q9 = q & p9 (--) q9 ) implies p (--) q )
proof
assume  p (--) q ; ::_thesis: ex i being Element of I ex p9, q9 being Element of (C . i) st
( p9 = p & q9 = q & p9 (--) q9 )
then  ex i being set ex P being pcs-Str ex p9, q9 being Element of P st
( i in I & P = C . i & p9 = p & q9 = q & p9 (--) q9 ) by Th12;
hence  ex i being Element of I ex p9, q9 being Element of (C . i) st
( p9 = p & q9 = q & p9 (--) q9 ) ; ::_thesis: verum
end;
thus  ( ex i being Element of I ex p9, q9 being Element of (C . i) st
( p9 = p & q9 = q & p9 (--) q9 ) implies p (--) q ) by Th12; ::_thesis: verum
end;

registration
let I be set ;
let C be reflexive-yielding () ManySortedSet of I;
cluster pcs-union C ->  reflexive strict ;
coherence
 pcs-union C is reflexive
proof
set P = pcs-union C;
set IR = the InternalRel of (pcs-union C);
set CP = the carrier of (pcs-union C);
set CA = Carrier C;
A1: the carrier of (pcs-union C) = Union (Carrier C) by Def36;
A2: the InternalRel of (pcs-union C) = Union (pcs-InternalRels C) by Def36;
A3: dom (pcs-InternalRels C) = I by PARTFUN1:def_2;
let x be set ; :: according to RELAT_2:def_1,ORDERS_2:def_2 ::_thesis: ( not x in the carrier of (pcs-union C) or [^,^] in the InternalRel of (pcs-union C) )
assume  x in the carrier of (pcs-union C) ; ::_thesis: [^,^] in the InternalRel of (pcs-union C)
then consider P being set  such that
A4: x in P and
A5: P in rng (Carrier C) by A1, TARSKI:def_4;
consider i being set  such that
A6: i in dom (Carrier C) and
A7: (Carrier C) . i = P by A5, FUNCT_1:def_3;
A8: ex R being 1-sorted st
( R = C . i & (Carrier C) . i = the carrier of R ) by A6, PRALG_1:def_13;
reconsider I = I as non empty set by A6;
reconsider i = i as Element of I by A6;
reconsider C = C as reflexive-yielding () ManySortedSet of I ;
A9: (pcs-InternalRels C) . i = the InternalRel of (C . i) by Def6;
 the InternalRel of (C . i) is_reflexive_in the carrier of (C . i) by ORDERS_2:def_2;
then A10: [x,x] in the InternalRel of (C . i) by A4, A7, A8, RELAT_2:def_1;
 the InternalRel of (C . i) in rng (pcs-InternalRels C) by A3, A9, FUNCT_1:3;
hence  [^,^] in the InternalRel of (pcs-union C) by A2, A10, TARSKI:def_4; ::_thesis: verum
end;
end;

registration
let I be set ;
let C be pcs-tol-reflexive-yielding () ManySortedSet of I;
cluster pcs-union C ->  pcs-tol-reflexive strict ;
coherence
 pcs-union C is pcs-tol-reflexive
proof
set P = pcs-union C;
set TR = the ToleranceRel of (pcs-union C);
set CP = the carrier of (pcs-union C);
set CA = Carrier C;
A1: the carrier of (pcs-union C) = Union (Carrier C) by Def36;
A2: the ToleranceRel of (pcs-union C) = Union (pcs-ToleranceRels C) by Def36;
A3: dom (pcs-ToleranceRels C) = I by PARTFUN1:def_2;
let x be set ; :: according to RELAT_2:def_1,PCS_0:def_9 ::_thesis: ( not x in the carrier of (pcs-union C) or [^,^] in the ToleranceRel of (pcs-union C) )
assume  x in the carrier of (pcs-union C) ; ::_thesis: [^,^] in the ToleranceRel of (pcs-union C)
then consider P being set  such that
A4: x in P and
A5: P in rng (Carrier C) by A1, TARSKI:def_4;
consider i being set  such that
A6: i in dom (Carrier C) and
A7: (Carrier C) . i = P by A5, FUNCT_1:def_3;
A8: ex R being 1-sorted st
( R = C . i & (Carrier C) . i = the carrier of R ) by A6, PRALG_1:def_13;
reconsider I = I as non empty set by A6;
reconsider i = i as Element of I by A6;
reconsider C = C as pcs-tol-reflexive-yielding () ManySortedSet of I ;
A9: (pcs-ToleranceRels C) . i = the ToleranceRel of (C . i) by Def20;
 the ToleranceRel of (C . i) is_reflexive_in the carrier of (C . i) by Def9;
then A10: [x,x] in the ToleranceRel of (C . i) by A4, A7, A8, RELAT_2:def_1;
 the ToleranceRel of (C . i) in rng (pcs-ToleranceRels C) by A3, A9, FUNCT_1:3;
hence  [^,^] in the ToleranceRel of (pcs-union C) by A2, A10, TARSKI:def_4; ::_thesis: verum
end;
end;

registration
let I be set ;
let C be pcs-tol-symmetric-yielding () ManySortedSet of I;
cluster pcs-union C ->  pcs-tol-symmetric strict ;
coherence
 pcs-union C is pcs-tol-symmetric
proof
set P = pcs-union C;
set TR = the ToleranceRel of (pcs-union C);
set CP = the carrier of (pcs-union C);
A1: the ToleranceRel of (pcs-union C) = Union (pcs-ToleranceRels C) by Def36;
let x, y be set ; :: according to RELAT_2:def_3,PCS_0:def_11 ::_thesis: ( not x in the carrier of (pcs-union C) or not y in the carrier of (pcs-union C) or not [^,^] in the ToleranceRel of (pcs-union C) or [^,^] in the ToleranceRel of (pcs-union C) )
assume that
 x in the carrier of (pcs-union C) and
 y in the carrier of (pcs-union C) ; ::_thesis: ( not [^,^] in the ToleranceRel of (pcs-union C) or [^,^] in the ToleranceRel of (pcs-union C) )
assume  [x,y] in the ToleranceRel of (pcs-union C) ; ::_thesis: [^,^] in the ToleranceRel of (pcs-union C)
then consider P being set  such that
A2: [x,y] in P and
A3: P in rng (pcs-ToleranceRels C) by A1, TARSKI:def_4;
consider i being set  such that
A4: i in dom (pcs-ToleranceRels C) and
A5: (pcs-ToleranceRels C) . i = P by A3, FUNCT_1:def_3;
reconsider I = I as non empty set by A4;
reconsider C = C as pcs-tol-symmetric-yielding () ManySortedSet of I ;
reconsider i = i as Element of I by A4;
A6: (pcs-ToleranceRels C) . i = the ToleranceRel of (C . i) by Def20;
then A7: x in the carrier of (C . i) by A2, A5, ZFMISC_1:87;
A8: y in the carrier of (C . i) by A2, A5, A6, ZFMISC_1:87;
 the ToleranceRel of (C . i) is_symmetric_in the carrier of (C . i) by Def11;
then A9: [y,x] in the ToleranceRel of (C . i) by A2, A5, A6, A7, A8, RELAT_2:def_3;
 the ToleranceRel of (C . i) in rng (pcs-ToleranceRels C) by A4, A6, FUNCT_1:3;
hence  [^,^] in the ToleranceRel of (pcs-union C) by A1, A9, TARSKI:def_4; ::_thesis: verum
end;
end;

registration
let I be set ;
let C be pcs-Chain of I;
cluster pcs-union C ->  transitive strict pcs-compatible ;
coherence
 ( pcs-union C is transitive & pcs-union C is pcs-compatible )
proof
set P = pcs-union C;
set IR = the InternalRel of (pcs-union C);
set TR = the ToleranceRel of (pcs-union C);
set CA = the carrier of (pcs-union C);
A1: the InternalRel of (pcs-union C) = Union (pcs-InternalRels C) by Def36;
A2: the ToleranceRel of (pcs-union C) = Union (pcs-ToleranceRels C) by Def36;
A3: dom C = I by PARTFUN1:def_2;
thus  pcs-union C is transitive ::_thesis: pcs-union C is pcs-compatible
proof
let x, y, z be set ; :: according to RELAT_2:def_8,ORDERS_2:def_3 ::_thesis: ( not x in the carrier of (pcs-union C) or not y in the carrier of (pcs-union C) or not z in the carrier of (pcs-union C) or not [^,^] in the InternalRel of (pcs-union C) or not [^,^] in the InternalRel of (pcs-union C) or [^,^] in the InternalRel of (pcs-union C) )
assume that
 x in the carrier of (pcs-union C) and
 y in the carrier of (pcs-union C) and
 z in the carrier of (pcs-union C) ; ::_thesis: ( not [^,^] in the InternalRel of (pcs-union C) or not [^,^] in the InternalRel of (pcs-union C) or [^,^] in the InternalRel of (pcs-union C) )
assume  [x,y] in the InternalRel of (pcs-union C) ; ::_thesis: ( not [^,^] in the InternalRel of (pcs-union C) or [^,^] in the InternalRel of (pcs-union C) )
then consider Z1 being set  such that
A4: [x,y] in Z1 and
A5: Z1 in rng (pcs-InternalRels C) by A1, TARSKI:def_4;
consider i being set  such that
A6: i in dom (pcs-InternalRels C) and
A7: (pcs-InternalRels C) . i = Z1 by A5, FUNCT_1:def_3;
assume  [y,z] in the InternalRel of (pcs-union C) ; ::_thesis: [^,^] in the InternalRel of (pcs-union C)
then consider Z2 being set  such that
A8: [y,z] in Z2 and
A9: Z2 in rng (pcs-InternalRels C) by A1, TARSKI:def_4;
consider j being set  such that
A10: j in dom (pcs-InternalRels C) and
A11: (pcs-InternalRels C) . j = Z2 by A9, FUNCT_1:def_3;
reconsider I = I as non empty set by A6;
reconsider C = C as pcs-Chain of I ;
reconsider i = i, j = j as Element of I by A6, A10;
A12: (pcs-InternalRels C) . i = the InternalRel of (C . i) by Def6;
then A13: x in the carrier of (C . i) by A4, A7, ZFMISC_1:87;
A14: y in the carrier of (C . i) by A4, A7, A12, ZFMISC_1:87;
A15: (pcs-InternalRels C) . j = the InternalRel of (C . j) by Def6;
A16: C . i in rng C by A3, FUNCT_1:3;
A17: C . j in rng C by A3, FUNCT_1:3;
A18: the InternalRel of (C . i) is_transitive_in the carrier of (C . i) by ORDERS_2:def_3;
A19: the InternalRel of (C . j) is_transitive_in the carrier of (C . j) by ORDERS_2:def_3;
percases ( C . i c= C . j or C . j c= C . i ) by A16, A17, Def35;
suppose C . i c= C . j ; ::_thesis: [^,^] in the InternalRel of (pcs-union C)
then A20: the InternalRel of (C . i) c= the InternalRel of (C . j) by Def34;
then A21: [x,y] in the InternalRel of (C . j) by A4, A7, A12;
then A22: x in the carrier of (C . j) by ZFMISC_1:87;
A23: y in the carrier of (C . j) by A21, ZFMISC_1:87;
 z in the carrier of (C . j) by A8, A11, A15, ZFMISC_1:87;
then A24: [x,z] in the InternalRel of (C . j) by A4, A7, A8, A11, A12, A15, A19, A20, A22, A23, RELAT_2:def_8;
 the InternalRel of (C . j) c= the InternalRel of (pcs-union C) by A1, A9, A11, A15, ZFMISC_1:74;
hence  [^,^] in the InternalRel of (pcs-union C) by A24; ::_thesis: verum
end;
suppose C . j c= C . i ; ::_thesis: [^,^] in the InternalRel of (pcs-union C)
then A25: the InternalRel of (C . j) c= the InternalRel of (C . i) by Def34;
then  [y,z] in the InternalRel of (C . i) by A8, A11, A15;
then  z in the carrier of (C . i) by ZFMISC_1:87;
then A26: [x,z] in the InternalRel of (C . i) by A4, A7, A8, A11, A12, A13, A14, A15, A18, A25, RELAT_2:def_8;
 the InternalRel of (C . i) c= the InternalRel of (pcs-union C) by A1, A5, A7, A12, ZFMISC_1:74;
hence  [^,^] in the InternalRel of (pcs-union C) by A26; ::_thesis: verum
end;
end;
end;
let p, p9, q, q9 be Element of (pcs-union C); :: according to PCS_0:def_22 ::_thesis: ( p (--) q & p9 <= p & q9 <= q implies p9 (--) q9 )
assume that
A27: p (--) q and
A28: p9 <= p and
A29: q9 <= q ; ::_thesis: p9 (--) q9
 [p9,p] in the InternalRel of (pcs-union C) by A28, ORDERS_2:def_5;
then consider Z1 being set  such that
A30: [p9,p] in Z1 and
A31: Z1 in rng (pcs-InternalRels C) by A1, TARSKI:def_4;
consider i being set  such that
A32: i in dom (pcs-InternalRels C) and
A33: (pcs-InternalRels C) . i = Z1 by A31, FUNCT_1:def_3;
reconsider I = I as non empty set by A32;
reconsider C = C as pcs-Chain of I ;
reconsider i = i as Element of I by A32;
A34: (pcs-ToleranceRels C) . i = the ToleranceRel of (C . i) by Def20;
A35: (pcs-InternalRels C) . i = the InternalRel of (C . i) by Def6;
then reconsider pi1 = p, p9i = p9 as Element of (C . i) by A30, A33, ZFMISC_1:87;
 [q9,q] in the InternalRel of (pcs-union C) by A29, ORDERS_2:def_5;
then consider Z2 being set  such that
A36: [q9,q] in Z2 and
A37: Z2 in rng (pcs-InternalRels C) by A1, TARSKI:def_4;
consider j being set  such that
A38: j in dom (pcs-InternalRels C) and
A39: (pcs-InternalRels C) . j = Z2 by A37, FUNCT_1:def_3;
reconsider j = j as Element of I by A38;
A40: (pcs-ToleranceRels C) . j = the ToleranceRel of (C . j) by Def20;
A41: (pcs-InternalRels C) . j = the InternalRel of (C . j) by Def6;
then A42: q9 in the carrier of (C . j) by A36, A39, ZFMISC_1:87;
A43: q in the carrier of (C . j) by A36, A39, A41, ZFMISC_1:87;
reconsider qj = q as Element of (C . j) by A36, A39, A41, ZFMISC_1:87;
 [p,q] in the ToleranceRel of (pcs-union C) by A27, Def7;
then consider Z3 being set  such that
A44: [p,q] in Z3 and
A45: Z3 in rng (pcs-ToleranceRels C) by A2, TARSKI:def_4;
consider k being set  such that
A46: k in dom (pcs-ToleranceRels C) and
A47: (pcs-ToleranceRels C) . k = Z3 by A45, FUNCT_1:def_3;
reconsider k = k as Element of I by A46;
A48: (pcs-ToleranceRels C) . k = the ToleranceRel of (C . k) by Def20;
then reconsider pk = p, qk = q as Element of (C . k) by A44, A47, ZFMISC_1:87;
A49: C . i in rng C by A3, FUNCT_1:3;
A50: C . j in rng C by A3, FUNCT_1:3;
A51: C . k in rng C by A3, FUNCT_1:3;
A52: dom (pcs-ToleranceRels C) = I by PARTFUN1:def_2;
then A53: the ToleranceRel of (C . i) c= the ToleranceRel of (pcs-union C) by A2, A34, FUNCT_1:3, ZFMISC_1:74;
A54: the ToleranceRel of (C . j) c= the ToleranceRel of (pcs-union C) by A2, A40, A52, FUNCT_1:3, ZFMISC_1:74;
A55: the ToleranceRel of (C . k) c= the ToleranceRel of (pcs-union C) by A2, A45, A47, A48, ZFMISC_1:74;
percases ( ( C . i c= C . j & C . j c= C . k ) or ( C . j c= C . i & C . i c= C . k ) or ( C . i c= C . k & C . k c= C . j ) or ( C . k c= C . i & C . i c= C . j ) or ( C . k c= C . j & C . j c= C . i ) or ( C . j c= C . k & C . k c= C . i ) ) by A49, A50, A51, Def35;
supposethat A56: C . i c= C . j and
A57: C . j c= C . k ; ::_thesis: p9 (--) q9
A58: the InternalRel of (C . j) c= the InternalRel of (C . k) by A57, Def34;
 the InternalRel of (C . i) c= the InternalRel of (C . j) by A56, Def34;
then A59: [p9,p] in the InternalRel of (C . j) by A30, A33, A35;
then  [p9,p] in the InternalRel of (C . k) by A58;
then reconsider p9k = p9 as Element of (C . k) by ZFMISC_1:87;
 [q9,q] in the InternalRel of (C . k) by A36, A39, A41, A58;
then reconsider q9k = q9 as Element of (C . k) by ZFMISC_1:87;
A60: p9k <= pk by A58, A59, ORDERS_2:def_5;
A61: q9k <= qk by A36, A39, A41, A58, ORDERS_2:def_5;
 pk (--) qk by A44, A47, A48, Def7;
then  p9k (--) q9k by A60, A61, Def22;
then  [p9k,q9k] in the ToleranceRel of (C . k) by Def7;
hence  [p9,q9] in the ToleranceRel of (pcs-union C) by A55; :: according to PCS_0:def_7 ::_thesis: verum
end;
supposethat A62: C . j c= C . i and
A63: C . i c= C . k ; ::_thesis: p9 (--) q9
A64: the InternalRel of (C . i) c= the InternalRel of (C . k) by A63, Def34;
A65: the InternalRel of (C . j) c= the InternalRel of (C . i) by A62, Def34;
 [p9,p] in the InternalRel of (C . k) by A30, A33, A35, A64;
then reconsider p9k = p9 as Element of (C . k) by ZFMISC_1:87;
A66: [q9,q] in the InternalRel of (C . i) by A36, A39, A41, A65;
then  [q9,q] in the InternalRel of (C . k) by A64;
then reconsider q9k = q9 as Element of (C . k) by ZFMISC_1:87;
A67: p9k <= pk by A30, A33, A35, A64, ORDERS_2:def_5;
A68: q9k <= qk by A64, A66, ORDERS_2:def_5;
 pk (--) qk by A44, A47, A48, Def7;
then  p9k (--) q9k by A67, A68, Def22;
then  [p9k,q9k] in the ToleranceRel of (C . k) by Def7;
hence  [p9,q9] in the ToleranceRel of (pcs-union C) by A55; :: according to PCS_0:def_7 ::_thesis: verum
end;
supposethat A69: C . i c= C . k and
A70: C . k c= C . j ; ::_thesis: p9 (--) q9
A71: the InternalRel of (C . k) c= the InternalRel of (C . j) by A70, Def34;
A72: the ToleranceRel of (C . k) c= the ToleranceRel of (C . j) by A70, Def34;
 the InternalRel of (C . i) c= the InternalRel of (C . k) by A69, Def34;
then A73: [p9,p] in the InternalRel of (C . k) by A30, A33, A35;
then A74: [p9,p] in the InternalRel of (C . j) by A71;
then reconsider p9j = p9 as Element of (C . j) by ZFMISC_1:87;
reconsider q9j = q9 as Element of (C . j) by A36, A39, A41, ZFMISC_1:87;
reconsider pj = p as Element of (C . j) by A74, ZFMISC_1:87;
A75: p9j <= pj by A71, A73, ORDERS_2:def_5;
A76: q9j <= qj by A36, A39, A41, ORDERS_2:def_5;
 pj (--) qj by A44, A47, A48, A72, Def7;
then  p9j (--) q9j by A75, A76, Def22;
then  [p9j,q9j] in the ToleranceRel of (C . j) by Def7;
hence  [p9,q9] in the ToleranceRel of (pcs-union C) by A54; :: according to PCS_0:def_7 ::_thesis: verum
end;
supposethat A77: C . k c= C . i and
A78: C . i c= C . j ; ::_thesis: p9 (--) q9
A79: the InternalRel of (C . i) c= the InternalRel of (C . j) by A78, Def34;
A80: the ToleranceRel of (C . i) c= the ToleranceRel of (C . j) by A78, Def34;
A81: the ToleranceRel of (C . k) c= the ToleranceRel of (C . i) by A77, Def34;
A82: [p9,p] in the InternalRel of (C . j) by A30, A33, A35, A79;
then reconsider p9j = p9 as Element of (C . j) by ZFMISC_1:87;
reconsider q9j = q9 as Element of (C . j) by A36, A39, A41, ZFMISC_1:87;
reconsider pj = p as Element of (C . j) by A82, ZFMISC_1:87;
 q in the carrier of (C . k) by A44, A47, A48, ZFMISC_1:87;
then reconsider qi = q as Element of (C . i) by A77, Lm2;
A83: p9j <= pj by A30, A33, A35, A79, ORDERS_2:def_5;
A84: q9j <= qj by A36, A39, A41, ORDERS_2:def_5;
 pi1 (--) qi by A44, A47, A48, A81, Def7;
then  pj (--) qj by A80, Th9;
then  p9j (--) q9j by A83, A84, Def22;
then  [p9j,q9j] in the ToleranceRel of (C . j) by Def7;
hence  [p9,q9] in the ToleranceRel of (pcs-union C) by A54; :: according to PCS_0:def_7 ::_thesis: verum
end;
supposethat A85: C . k c= C . j and
A86: C . j c= C . i ; ::_thesis: p9 (--) q9
A87: the ToleranceRel of (C . j) c= the ToleranceRel of (C . i) by A86, Def34;
A88: the ToleranceRel of (C . k) c= the ToleranceRel of (C . j) by A85, Def34;
A89: the InternalRel of (C . j) c= the InternalRel of (C . i) by A86, Def34;
reconsider q9i = q9 as Element of (C . i) by A42, A86, Lm2;
reconsider qi = q as Element of (C . i) by A43, A86, Lm2;
 p in the carrier of (C . k) by A44, A47, A48, ZFMISC_1:87;
then reconsider pj = p as Element of (C . j) by A85, Lm2;
A90: p9i <= pi1 by A30, A33, A35, ORDERS_2:def_5;
A91: q9i <= qi by A36, A39, A41, A89, ORDERS_2:def_5;
 pj (--) qj by A44, A47, A48, A88, Def7;
then  pi1 (--) qi by A87, Th9;
then  p9i (--) q9i by A90, A91, Def22;
then  [p9i,q9i] in the ToleranceRel of (C . i) by Def7;
hence  [p9,q9] in the ToleranceRel of (pcs-union C) by A53; :: according to PCS_0:def_7 ::_thesis: verum
end;
supposethat A92: C . j c= C . k and
A93: C . k c= C . i ; ::_thesis: p9 (--) q9
A94: the ToleranceRel of (C . k) c= the ToleranceRel of (C . i) by A93, Def34;
A95: the InternalRel of (C . k) c= the InternalRel of (C . i) by A93, Def34;
A96: the InternalRel of (C . j) c= the InternalRel of (C . k) by A92, Def34;
reconsider q9k = q9 as Element of (C . k) by A42, A92, Lm2;
A97: the carrier of (C . j) c= the carrier of (C . k) by A92, Def34;
then reconsider q9i = q9 as Element of (C . i) by A42, A93, Lm2;
reconsider qi = q as Element of (C . i) by A43, A93, A97, Lm2;
A98: q9k <= qk by A36, A39, A41, A96, ORDERS_2:def_5;
A99: p9i <= pi1 by A30, A33, A35, ORDERS_2:def_5;
A100: q9i <= qi by A95, A98, Th8;
 pi1 (--) qi by A44, A47, A48, A94, Def7;
then  p9i (--) q9i by A99, A100, Def22;
then  [p9i,q9i] in the ToleranceRel of (C . i) by Def7;
hence  [p9,q9] in the ToleranceRel of (pcs-union C) by A53; :: according to PCS_0:def_7 ::_thesis: verum
end;
end;
end;
end;

registration
let p, q be set ;
cluster<%p,q%> -> {0,1} -defined ;
coherence
 <%p,q%> is {0,1} -defined
proof
 len <%p,q%> = {0,1} by AFINSQ_1:38, CARD_1:50;
hence  <%p,q%> is {0,1} -defined by RELAT_1:def_18; ::_thesis: verum
end;
cluster<%p,q%> ->  total ;
coherence
 <%p,q%> is total
proof
 len <%p,q%> = {0,1} by AFINSQ_1:38, CARD_1:50;
hence  <%p,q%> is total by PARTFUN1:def_2; ::_thesis: verum
end;
end;

registration
let P, Q be 1-sorted ;
cluster<%P,Q%> ->  1-sorted-yielding ;
coherence
 <%P,Q%> is 1-sorted-yielding
proof
let x be set ; :: according to PRALG_1:def_11 ::_thesis: ( not x in proj1 <%P,Q%> or <%P,Q%> . x is 1-sorted )
assume  x in dom <%P,Q%> ; ::_thesis: <%P,Q%> . x is 1-sorted
then  x in len <%P,Q%> ;
then  x in 2 by AFINSQ_1:38;
then  ( x = 0 or x = 1 ) by CARD_1:50, TARSKI:def_2;
hence  <%P,Q%> . x is 1-sorted by AFINSQ_1:38; ::_thesis: verum
end;
end;

Lm3: now__::_thesis:_for_a,_b_being_set_holds_rng_<%a,b%>_=_{a,b}
let a, b be set ; ::_thesis: rng <%a,b%> = {a,b}
 <%a,b%> = (0,1) --> (a,b) by AFINSQ_1:76;
hence  rng <%a,b%> = {a,b} by FUNCT_4:64; ::_thesis: verum
end;

registration
let P, Q be RelStr ;
cluster<%P,Q%> ->  RelStr-yielding ;
coherence
 <%P,Q%> is RelStr-yielding
proof
let x be set ; :: according to YELLOW_1:def_3 ::_thesis: ( not x in proj2 <%P,Q%> or x is RelStr )
assume  x in rng <%P,Q%> ; ::_thesis: x is RelStr
then  x in {P,Q} by Lm3;
hence  x is RelStr by TARSKI:def_2; ::_thesis: verum
end;
end;

registration
let P, Q be TolStr ;
cluster<%P,Q%> -> () ;
coherence
 <%P,Q%> is TolStr-yielding
proof
let x be set ; :: according to PCS_0:def_15 ::_thesis: ( x in {0,1} implies <%P,Q%> . x is TolStr )
assume  x in {0,1} ; ::_thesis: <%P,Q%> . x is TolStr
then  ( x = 0 or x = 1 ) by TARSKI:def_2;
hence  <%P,Q%> . x is TolStr by AFINSQ_1:38; ::_thesis: verum
end;
end;

registration
let P, Q be pcs-Str ;
cluster<%P,Q%> -> () ;
coherence
 <%P,Q%> is pcs-Str-yielding
proof
let x be set ; :: according to PCS_0:def_32 ::_thesis: ( x in {0,1} implies <%P,Q%> . x is pcs-Str )
assume  x in {0,1} ; ::_thesis: <%P,Q%> . x is pcs-Str
then  ( x = 0 or x = 1 ) by TARSKI:def_2;
hence  <%P,Q%> . x is pcs-Str by AFINSQ_1:38; ::_thesis: verum
end;
end;

registration
let P, Q be reflexive pcs-Str ;
cluster<%P,Q%> ->  reflexive-yielding ;
coherence
 <%P,Q%> is reflexive-yielding
proof
let x be RelStr ; :: according to WAYBEL_3:def_8 ::_thesis: ( not x in proj2 <%P,Q%> or x is reflexive )
assume  x in rng <%P,Q%> ; ::_thesis: x is reflexive
then  x in {P,Q} by Lm3;
hence  x is reflexive by TARSKI:def_2; ::_thesis: verum
end;
end;

registration
let P, Q be transitive pcs-Str ;
cluster<%P,Q%> ->  transitive-yielding ;
coherence
 <%P,Q%> is transitive-yielding
proof
let x be RelStr ; :: according to PCS_0:def_4 ::_thesis: ( x in rng <%P,Q%> implies x is transitive )
assume  x in rng <%P,Q%> ; ::_thesis: x is transitive
then  x in {P,Q} by Lm3;
hence  x is transitive by TARSKI:def_2; ::_thesis: verum
end;
end;

registration
let P, Q be pcs-tol-reflexive pcs-Str ;
cluster<%P,Q%> ->  pcs-tol-reflexive-yielding ;
coherence
 <%P,Q%> is pcs-tol-reflexive-yielding
proof
let x be TolStr ; :: according to PCS_0:def_16 ::_thesis: ( x in rng <%P,Q%> implies x is pcs-tol-reflexive )
assume  x in rng <%P,Q%> ; ::_thesis: x is pcs-tol-reflexive
then  x in {P,Q} by Lm3;
hence  x is pcs-tol-reflexive by TARSKI:def_2; ::_thesis: verum
end;
end;

registration
let P, Q be pcs-tol-symmetric pcs-Str ;
cluster<%P,Q%> ->  pcs-tol-symmetric-yielding ;
coherence
 <%P,Q%> is pcs-tol-symmetric-yielding
proof
let x be TolStr ; :: according to PCS_0:def_18 ::_thesis: ( x in rng <%P,Q%> implies x is pcs-tol-symmetric )
assume  x in rng <%P,Q%> ; ::_thesis: x is pcs-tol-symmetric
then  x in {P,Q} by Lm3;
hence  x is pcs-tol-symmetric by TARSKI:def_2; ::_thesis: verum
end;
end;

registration
let P, Q be pcs;
cluster<%P,Q%> -> () ;
coherence
 <%P,Q%> is pcs-yielding
proof
let x be set ; :: according to PCS_0:def_33 ::_thesis: ( x in {0,1} implies <%P,Q%> . x is pcs )
assume  x in {0,1} ; ::_thesis: <%P,Q%> . x is pcs
then  ( x = 0 or x = 1 ) by TARSKI:def_2;
hence  <%P,Q%> . x is pcs by AFINSQ_1:38; ::_thesis: verum
end;
end;

definition
canceled;
let P, Q be pcs-Str ;
func pcs-sum (P,Q) ->  pcs-Str  equals :: PCS_0:def 38
 pcs-union <%P,Q%>;
coherence
 pcs-union <%P,Q%> is pcs-Str ;
end;

:: deftheorem  PCS_0:def_37_:_
canceled;

:: deftheorem  defines pcs-sum PCS_0:def_38_:_
 for P, Q being pcs-Str holds pcs-sum (P,Q) = pcs-union <%P,Q%>;

deffunc H1( pcs-Str , pcs-Str ) ->  pcs-Str = pcs-Str(# ( the carrier of $1 \/ the carrier of $2),( the InternalRel of $1 \/ the InternalRel of $2),( the ToleranceRel of $1 \/ the ToleranceRel of $2) #);

theorem Th14: :: PCS_0:14
for P, Q being pcs-Str holds
( the carrier of (pcs-sum (P,Q)) = the carrier of P \/ the carrier of Q & the InternalRel of (pcs-sum (P,Q)) = the InternalRel of P \/ the InternalRel of Q & the ToleranceRel of (pcs-sum (P,Q)) = the ToleranceRel of P \/ the ToleranceRel of Q )
proof
let P, Q be pcs-Str ; ::_thesis: ( the carrier of (pcs-sum (P,Q)) = the carrier of P \/ the carrier of Q & the InternalRel of (pcs-sum (P,Q)) = the InternalRel of P \/ the InternalRel of Q & the ToleranceRel of (pcs-sum (P,Q)) = the ToleranceRel of P \/ the ToleranceRel of Q )
set S = H1(P,Q);
set f = <%P,Q%>;
A1: dom (Carrier <%P,Q%>) = {0,1} by PARTFUN1:def_2;
A2: dom (pcs-InternalRels <%P,Q%>) = {0,1} by PARTFUN1:def_2;
A3: dom (pcs-ToleranceRels <%P,Q%>) = {0,1} by PARTFUN1:def_2;
A4: <%P,Q%> . 0 = P by AFINSQ_1:38;
A5: <%P,Q%> . 1 = Q by AFINSQ_1:38;
A6: the carrier of H1(P,Q) = Union (Carrier <%P,Q%>)
proof
thus  the carrier of H1(P,Q) c= Union (Carrier <%P,Q%>) :: according to XBOOLE_0:def_10 ::_thesis: Union (Carrier <%P,Q%>) c= the carrier of H1(P,Q)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of H1(P,Q) or x in Union (Carrier <%P,Q%>) )
assume  x in the carrier of H1(P,Q) ; ::_thesis: x in Union (Carrier <%P,Q%>)
then A7: ( x in the carrier of P or x in the carrier of Q ) by XBOOLE_0:def_3;
A8: (Carrier <%P,Q%>) . z = the carrier of (<%P,Q%> . z) by Def1;
A9: (Carrier <%P,Q%>) . j = the carrier of (<%P,Q%> . j) by Def1;
A10: the carrier of P in rng (Carrier <%P,Q%>) by A1, A4, A8, FUNCT_1:3;
 the carrier of Q in rng (Carrier <%P,Q%>) by A1, A5, A9, FUNCT_1:3;
hence  x in Union (Carrier <%P,Q%>) by A7, A10, TARSKI:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Union (Carrier <%P,Q%>) or x in the carrier of H1(P,Q) )
assume  x in Union (Carrier <%P,Q%>) ; ::_thesis: x in the carrier of H1(P,Q)
then consider Z being set  such that
A11: x in Z and
A12: Z in rng (Carrier <%P,Q%>) by TARSKI:def_4;
consider i being set  such that
A13: i in dom (Carrier <%P,Q%>) and
A14: (Carrier <%P,Q%>) . i = Z by A12, FUNCT_1:def_3;
 ( i = 0 or i = 1 ) by A13, TARSKI:def_2;
then  ( Z = the carrier of (<%P,Q%> . z) or Z = the carrier of (<%P,Q%> . j) ) by A14, Def1;
hence  x in the carrier of H1(P,Q) by A4, A5, A11, XBOOLE_0:def_3; ::_thesis: verum
end;
A15: the InternalRel of H1(P,Q) = Union (pcs-InternalRels <%P,Q%>)
proof
thus  the InternalRel of H1(P,Q) c= Union (pcs-InternalRels <%P,Q%>) :: according to XBOOLE_0:def_10 ::_thesis: Union (pcs-InternalRels <%P,Q%>) c= the InternalRel of H1(P,Q)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the InternalRel of H1(P,Q) or x in Union (pcs-InternalRels <%P,Q%>) )
assume  x in the InternalRel of H1(P,Q) ; ::_thesis: x in Union (pcs-InternalRels <%P,Q%>)
then A16: ( x in the InternalRel of P or x in the InternalRel of Q ) by XBOOLE_0:def_3;
A17: (pcs-InternalRels <%P,Q%>) . z = the InternalRel of (<%P,Q%> . z) by Def6;
A18: (pcs-InternalRels <%P,Q%>) . j = the InternalRel of (<%P,Q%> . j) by Def6;
A19: the InternalRel of P in rng (pcs-InternalRels <%P,Q%>) by A2, A4, A17, FUNCT_1:3;
 the InternalRel of Q in rng (pcs-InternalRels <%P,Q%>) by A2, A5, A18, FUNCT_1:3;
hence  x in Union (pcs-InternalRels <%P,Q%>) by A16, A19, TARSKI:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Union (pcs-InternalRels <%P,Q%>) or x in the InternalRel of H1(P,Q) )
assume  x in Union (pcs-InternalRels <%P,Q%>) ; ::_thesis: x in the InternalRel of H1(P,Q)
then consider Z being set  such that
A20: x in Z and
A21: Z in rng (pcs-InternalRels <%P,Q%>) by TARSKI:def_4;
consider i being set  such that
A22: i in dom (pcs-InternalRels <%P,Q%>) and
A23: (pcs-InternalRels <%P,Q%>) . i = Z by A21, FUNCT_1:def_3;
 ( i = 0 or i = 1 ) by A22, TARSKI:def_2;
then  ( Z = the InternalRel of (<%P,Q%> . z) or Z = the InternalRel of (<%P,Q%> . j) ) by A23, Def6;
hence  x in the InternalRel of H1(P,Q) by A4, A5, A20, XBOOLE_0:def_3; ::_thesis: verum
end;
 the ToleranceRel of H1(P,Q) = Union (pcs-ToleranceRels <%P,Q%>)
proof
thus  the ToleranceRel of H1(P,Q) c= Union (pcs-ToleranceRels <%P,Q%>) :: according to XBOOLE_0:def_10 ::_thesis: Union (pcs-ToleranceRels <%P,Q%>) c= the ToleranceRel of H1(P,Q)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the ToleranceRel of H1(P,Q) or x in Union (pcs-ToleranceRels <%P,Q%>) )
assume  x in the ToleranceRel of H1(P,Q) ; ::_thesis: x in Union (pcs-ToleranceRels <%P,Q%>)
then A24: ( x in the ToleranceRel of P or x in the ToleranceRel of Q ) by XBOOLE_0:def_3;
A25: (pcs-ToleranceRels <%P,Q%>) . z = the ToleranceRel of (<%P,Q%> . z) by Def20;
A26: (pcs-ToleranceRels <%P,Q%>) . j = the ToleranceRel of (<%P,Q%> . j) by Def20;
A27: the ToleranceRel of P in rng (pcs-ToleranceRels <%P,Q%>) by A3, A4, A25, FUNCT_1:3;
 the ToleranceRel of Q in rng (pcs-ToleranceRels <%P,Q%>) by A3, A5, A26, FUNCT_1:3;
hence  x in Union (pcs-ToleranceRels <%P,Q%>) by A24, A27, TARSKI:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Union (pcs-ToleranceRels <%P,Q%>) or x in the ToleranceRel of H1(P,Q) )
assume  x in Union (pcs-ToleranceRels <%P,Q%>) ; ::_thesis: x in the ToleranceRel of H1(P,Q)
then consider Z being set  such that
A28: x in Z and
A29: Z in rng (pcs-ToleranceRels <%P,Q%>) by TARSKI:def_4;
consider i being set  such that
A30: i in dom (pcs-ToleranceRels <%P,Q%>) and
A31: (pcs-ToleranceRels <%P,Q%>) . i = Z by A29, FUNCT_1:def_3;
 ( i = 0 or i = 1 ) by A30, TARSKI:def_2;
then  ( Z = the ToleranceRel of (<%P,Q%> . z) or Z = the ToleranceRel of (<%P,Q%> . j) ) by A31, Def20;
hence  x in the ToleranceRel of H1(P,Q) by A4, A5, A28, XBOOLE_0:def_3; ::_thesis: verum
end;
hence  ( the carrier of (pcs-sum (P,Q)) = the carrier of P \/ the carrier of Q & the InternalRel of (pcs-sum (P,Q)) = the InternalRel of P \/ the InternalRel of Q & the ToleranceRel of (pcs-sum (P,Q)) = the ToleranceRel of P \/ the ToleranceRel of Q ) by A6, A15, Def36; ::_thesis: verum
end;

theorem Th15: :: PCS_0:15
for P, Q being pcs-Str holds pcs-sum (P,Q) = pcs-Str(# ( the carrier of P \/ the carrier of Q),( the InternalRel of P \/ the InternalRel of Q),( the ToleranceRel of P \/ the ToleranceRel of Q) #)
proof
let P, Q be pcs-Str ; ::_thesis: pcs-sum (P,Q) = pcs-Str(# ( the carrier of P \/ the carrier of Q),( the InternalRel of P \/ the InternalRel of Q),( the ToleranceRel of P \/ the ToleranceRel of Q) #)
A1: the carrier of (pcs-sum (P,Q)) = the carrier of P \/ the carrier of Q by Th14;
 the InternalRel of (pcs-sum (P,Q)) = the InternalRel of P \/ the InternalRel of Q by Th14;
hence  pcs-sum (P,Q) = pcs-Str(# ( the carrier of P \/ the carrier of Q),( the InternalRel of P \/ the InternalRel of Q),( the ToleranceRel of P \/ the ToleranceRel of Q) #) by A1, Th14; ::_thesis: verum
end;

theorem :: PCS_0:16
for P, Q being pcs-Str
for p, q being Element of (pcs-sum (P,Q)) holds
( p <= q iff ( ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 <= q9 ) or ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 <= q9 ) ) )
proof
let P, Q be pcs-Str ; ::_thesis: for p, q being Element of (pcs-sum (P,Q)) holds
( p <= q iff ( ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 <= q9 ) or ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 <= q9 ) ) )
set R = pcs-sum (P,Q);
let p, q be Element of (pcs-sum (P,Q)); ::_thesis: ( p <= q iff ( ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 <= q9 ) or ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 <= q9 ) ) )
A1: the InternalRel of (pcs-sum (P,Q)) = the InternalRel of P \/ the InternalRel of Q by Th14;
thus  ( not p <= q or ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 <= q9 ) or ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 <= q9 ) ) ::_thesis: ( ( ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 <= q9 ) or ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 <= q9 ) ) implies p <= q )
proof
assume A2: [p,q] in the InternalRel of (pcs-sum (P,Q)) ; :: according to ORDERS_2:def_5 ::_thesis: ( ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 <= q9 ) or ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 <= q9 ) )
percases ( [p,q] in the InternalRel of P or [p,q] in the InternalRel of Q ) by A1, A2, XBOOLE_0:def_3;
supposeA3: [p,q] in the InternalRel of P ; ::_thesis: ( ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 <= q9 ) or ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 <= q9 ) )
then reconsider p9 = p, q9 = q as Element of P by ZFMISC_1:87;
 p9 <= q9 by A3, ORDERS_2:def_5;
hence  ( ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 <= q9 ) or ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 <= q9 ) ) ; ::_thesis: verum
end;
supposeA4: [p,q] in the InternalRel of Q ; ::_thesis: ( ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 <= q9 ) or ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 <= q9 ) )
then reconsider p9 = p, q9 = q as Element of Q by ZFMISC_1:87;
 p9 <= q9 by A4, ORDERS_2:def_5;
hence  ( ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 <= q9 ) or ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 <= q9 ) ) ; ::_thesis: verum
end;
end;
end;
assume A5: ( ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 <= q9 ) or ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 <= q9 ) ) ; ::_thesis: p <= q
percases ( ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 <= q9 ) or ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 <= q9 ) ) by A5;
suppose ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 <= q9 ) ; ::_thesis: p <= q
then consider p9, q9 being Element of P such that
A6: p9 = p and
A7: q9 = q and
A8: p9 <= q9 ;
 [p9,q9] in the InternalRel of P by A8, ORDERS_2:def_5;
hence  [p,q] in the InternalRel of (pcs-sum (P,Q)) by A1, A6, A7, XBOOLE_0:def_3; :: according to ORDERS_2:def_5 ::_thesis: verum
end;
suppose ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 <= q9 ) ; ::_thesis: p <= q
then consider p9, q9 being Element of Q such that
A9: p9 = p and
A10: q9 = q and
A11: p9 <= q9 ;
 [p9,q9] in the InternalRel of Q by A11, ORDERS_2:def_5;
hence  [p,q] in the InternalRel of (pcs-sum (P,Q)) by A1, A9, A10, XBOOLE_0:def_3; :: according to ORDERS_2:def_5 ::_thesis: verum
end;
end;
end;

theorem :: PCS_0:17
for P, Q being pcs-Str
for p, q being Element of (pcs-sum (P,Q)) holds
( p (--) q iff ( ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 (--) q9 ) or ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 (--) q9 ) ) )
proof
let P, Q be pcs-Str ; ::_thesis: for p, q being Element of (pcs-sum (P,Q)) holds
( p (--) q iff ( ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 (--) q9 ) or ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 (--) q9 ) ) )
set R = pcs-sum (P,Q);
let p, q be Element of (pcs-sum (P,Q)); ::_thesis: ( p (--) q iff ( ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 (--) q9 ) or ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 (--) q9 ) ) )
A1: the ToleranceRel of (pcs-sum (P,Q)) = the ToleranceRel of P \/ the ToleranceRel of Q by Th14;
thus  ( not p (--) q or ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 (--) q9 ) or ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 (--) q9 ) ) ::_thesis: ( ( ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 (--) q9 ) or ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 (--) q9 ) ) implies p (--) q )
proof
assume A2: [p,q] in the ToleranceRel of (pcs-sum (P,Q)) ; :: according to PCS_0:def_7 ::_thesis: ( ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 (--) q9 ) or ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 (--) q9 ) )
percases ( [p,q] in the ToleranceRel of P or [p,q] in the ToleranceRel of Q ) by A1, A2, XBOOLE_0:def_3;
supposeA3: [p,q] in the ToleranceRel of P ; ::_thesis: ( ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 (--) q9 ) or ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 (--) q9 ) )
then reconsider p9 = p, q9 = q as Element of P by ZFMISC_1:87;
 p9 (--) q9 by A3, Def7;
hence  ( ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 (--) q9 ) or ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 (--) q9 ) ) ; ::_thesis: verum
end;
supposeA4: [p,q] in the ToleranceRel of Q ; ::_thesis: ( ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 (--) q9 ) or ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 (--) q9 ) )
then reconsider p9 = p, q9 = q as Element of Q by ZFMISC_1:87;
 p9 (--) q9 by A4, Def7;
hence  ( ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 (--) q9 ) or ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 (--) q9 ) ) ; ::_thesis: verum
end;
end;
end;
assume A5: ( ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 (--) q9 ) or ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 (--) q9 ) ) ; ::_thesis: p (--) q
percases ( ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 (--) q9 ) or ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 (--) q9 ) ) by A5;
suppose ex p9, q9 being Element of P st
( p9 = p & q9 = q & p9 (--) q9 ) ; ::_thesis: p (--) q
then consider p9, q9 being Element of P such that
A6: p9 = p and
A7: q9 = q and
A8: p9 (--) q9 ;
 [p9,q9] in the ToleranceRel of P by A8, Def7;
hence  [p,q] in the ToleranceRel of (pcs-sum (P,Q)) by A1, A6, A7, XBOOLE_0:def_3; :: according to PCS_0:def_7 ::_thesis: verum
end;
suppose ex p9, q9 being Element of Q st
( p9 = p & q9 = q & p9 (--) q9 ) ; ::_thesis: p (--) q
then consider p9, q9 being Element of Q such that
A9: p9 = p and
A10: q9 = q and
A11: p9 (--) q9 ;
 [p9,q9] in the ToleranceRel of Q by A11, Def7;
hence  [p,q] in the ToleranceRel of (pcs-sum (P,Q)) by A1, A9, A10, XBOOLE_0:def_3; :: according to PCS_0:def_7 ::_thesis: verum
end;
end;
end;

registration
let P, Q be reflexive pcs-Str ;
cluster pcs-sum (P,Q) ->  reflexive ;
coherence
 pcs-sum (P,Q) is reflexive ;
end;

registration
let P, Q be pcs-tol-reflexive pcs-Str ;
cluster pcs-sum (P,Q) ->  pcs-tol-reflexive ;
coherence
 pcs-sum (P,Q) is pcs-tol-reflexive ;
end;

registration
let P, Q be pcs-tol-symmetric pcs-Str ;
cluster pcs-sum (P,Q) ->  pcs-tol-symmetric ;
coherence
 pcs-sum (P,Q) is pcs-tol-symmetric ;
end;

theorem Th18: :: PCS_0:18
for P, Q being pcs st P misses Q holds
the InternalRel of (pcs-sum (P,Q)) is transitive
proof
let P, Q be pcs; ::_thesis: ( P misses Q implies the InternalRel of (pcs-sum (P,Q)) is transitive )
assume A1: the carrier of P misses the carrier of Q ; :: according to TSEP_1:def_3 ::_thesis: the InternalRel of (pcs-sum (P,Q)) is transitive
 pcs-sum (P,Q) = H1(P,Q) by Th15;
hence  the InternalRel of (pcs-sum (P,Q)) is transitive by A1, Th1; ::_thesis: verum
end;

theorem Th19: :: PCS_0:19
for P, Q being pcs st P misses Q holds
pcs-sum (P,Q) is pcs-compatible
proof
let P, Q be pcs; ::_thesis: ( P misses Q implies pcs-sum (P,Q) is pcs-compatible )
set D1 = the carrier of P;
set D2 = the carrier of Q;
set R1 = the InternalRel of P;
set R2 = the InternalRel of Q;
set T1 = the ToleranceRel of P;
set T2 = the ToleranceRel of Q;
set R = the InternalRel of P \/ the InternalRel of Q;
set T = the ToleranceRel of P \/ the ToleranceRel of Q;
assume A1: the carrier of P misses the carrier of Q ; :: according to TSEP_1:def_3 ::_thesis: pcs-sum (P,Q) is pcs-compatible
let p, p9, q, q9 be Element of (pcs-sum (P,Q)); :: according to PCS_0:def_22 ::_thesis: ( p (--) q & p9 <= p & q9 <= q implies p9 (--) q9 )
assume that
A2: p (--) q and
A3: p9 <= p and
A4: q9 <= q ; ::_thesis: p9 (--) q9
A5: pcs-sum (P,Q) = H1(P,Q) by Th15;
then A6: [p,q] in the ToleranceRel of P \/ the ToleranceRel of Q by A2, Def7;
percases ( [p,q] in the ToleranceRel of P or [p,q] in the ToleranceRel of Q ) by A6, XBOOLE_0:def_3;
supposeA7: [p,q] in the ToleranceRel of P ; ::_thesis: p9 (--) q9
then A8: p in the carrier of P by ZFMISC_1:87;
A9: q in the carrier of P by A7, ZFMISC_1:87;
reconsider p1 = p, q1 = q as Element of P by A7, ZFMISC_1:87;
A10: p1 (--) q1 by A7, Def7;
A11: [p9,p] in the InternalRel of P \/ the InternalRel of Q by A3, A5, ORDERS_2:def_5;
A12: [q9,q] in the InternalRel of P \/ the InternalRel of Q by A4, A5, ORDERS_2:def_5;
then reconsider p91 = p9, q91 = q9 as Element of P by A1, A8, A9, A11, Lm1;
A13: [p9,p] in the InternalRel of P by A1, A8, A11, Lm1;
A14: [q9,q] in the InternalRel of P by A1, A9, A12, Lm1;
A15: p91 <= p1 by A13, ORDERS_2:def_5;
 q91 <= q1 by A14, ORDERS_2:def_5;
then  p91 (--) q91 by A10, A15, Def22;
then  [p91,q91] in the ToleranceRel of P by Def7;
then  [p91,q91] in the ToleranceRel of P \/ the ToleranceRel of Q by XBOOLE_0:def_3;
hence  p9 (--) q9 by A5, Def7; ::_thesis: verum
end;
supposeA16: [p,q] in the ToleranceRel of Q ; ::_thesis: p9 (--) q9
then A17: p in the carrier of Q by ZFMISC_1:87;
A18: q in the carrier of Q by A16, ZFMISC_1:87;
reconsider p1 = p, q1 = q as Element of Q by A16, ZFMISC_1:87;
A19: p1 (--) q1 by A16, Def7;
A20: [p9,p] in the InternalRel of P \/ the InternalRel of Q by A3, A5, ORDERS_2:def_5;
A21: [q9,q] in the InternalRel of P \/ the InternalRel of Q by A4, A5, ORDERS_2:def_5;
then reconsider p91 = p9, q91 = q9 as Element of Q by A1, A17, A18, A20, Lm1;
A22: [p9,p] in the InternalRel of Q by A1, A17, A20, Lm1;
A23: [q9,q] in the InternalRel of Q by A1, A18, A21, Lm1;
A24: p91 <= p1 by A22, ORDERS_2:def_5;
 q91 <= q1 by A23, ORDERS_2:def_5;
then  p91 (--) q91 by A19, A24, Def22;
then  [p91,q91] in the ToleranceRel of Q by Def7;
then  [p91,q91] in the ToleranceRel of P \/ the ToleranceRel of Q by XBOOLE_0:def_3;
hence  p9 (--) q9 by A5, Def7; ::_thesis: verum
end;
end;
end;

theorem :: PCS_0:20
for P, Q being pcs st P misses Q holds
pcs-sum (P,Q) is pcs
proof
let P, Q be pcs; ::_thesis: ( P misses Q implies pcs-sum (P,Q) is pcs )
assume A1: P misses Q ; ::_thesis: pcs-sum (P,Q) is pcs
set R = pcs-sum (P,Q);
A2: field the InternalRel of (pcs-sum (P,Q)) = the carrier of (pcs-sum (P,Q)) by ORDERS_1:12;
 the InternalRel of (pcs-sum (P,Q)) is transitive by A1, Th18;
then  the InternalRel of (pcs-sum (P,Q)) is_transitive_in the carrier of (pcs-sum (P,Q)) by A2, RELAT_2:def_16;
then A3: pcs-sum (P,Q) is transitive by ORDERS_2:def_3;
 pcs-sum (P,Q) is pcs-compatible by A1, Th19;
hence  pcs-sum (P,Q) is pcs by A3; ::_thesis: verum
end;

definition
let P be pcs-Str ;
let a be set ;
func pcs-extension (P,a) ->  strict pcs-Str  means :Def39: :: PCS_0:def 39
( the carrier of it = {a} \/ the carrier of P & the InternalRel of it = [:{a}, the carrier of it:] \/ the InternalRel of P & the ToleranceRel of it = ([:{a}, the carrier of it:] \/ [: the carrier of it,{a}:]) \/ the ToleranceRel of P );
existence
 ex b1 being strict pcs-Str st
( the carrier of b1 = {a} \/ the carrier of P & the InternalRel of b1 = [:{a}, the carrier of b1:] \/ the InternalRel of P & the ToleranceRel of b1 = ([:{a}, the carrier of b1:] \/ [: the carrier of b1,{a}:]) \/ the ToleranceRel of P )
proof
set D = {a} \/ the carrier of P;
set IR = [:{a},({a} \/ the carrier of P):] \/ the InternalRel of P;
set TR = ([:({a} \/ the carrier of P),{a}:] \/ [:{a},({a} \/ the carrier of P):]) \/ the ToleranceRel of P;
A1: {a} c= {a} \/ the carrier of P by XBOOLE_1:7;
then A2: [:{a},({a} \/ the carrier of P):] c= [:({a} \/ the carrier of P),({a} \/ the carrier of P):] by ZFMISC_1:95;
 the carrier of P c= {a} \/ the carrier of P by XBOOLE_1:7;
then A3: [: the carrier of P, the carrier of P:] c= [:({a} \/ the carrier of P),({a} \/ the carrier of P):] by ZFMISC_1:96;
then  the InternalRel of P c= [:({a} \/ the carrier of P),({a} \/ the carrier of P):] by XBOOLE_1:1;
then reconsider IR = [:{a},({a} \/ the carrier of P):] \/ the InternalRel of P as Relation of ({a} \/ the carrier of P) by A2, XBOOLE_1:8;
 [:({a} \/ the carrier of P),{a}:] c= [:({a} \/ the carrier of P),({a} \/ the carrier of P):] by A1, ZFMISC_1:95;
then A4: [:({a} \/ the carrier of P),{a}:] \/ [:{a},({a} \/ the carrier of P):] c= [:({a} \/ the carrier of P),({a} \/ the carrier of P):] by A2, XBOOLE_1:8;
 the ToleranceRel of P c= [:({a} \/ the carrier of P),({a} \/ the carrier of P):] by A3, XBOOLE_1:1;
then reconsider TR = ([:({a} \/ the carrier of P),{a}:] \/ [:{a},({a} \/ the carrier of P):]) \/ the ToleranceRel of P as Relation of ({a} \/ the carrier of P) by A4, XBOOLE_1:8;
take  pcs-Str(# ({a} \/ the carrier of P),IR,TR #) ; ::_thesis: ( the carrier of pcs-Str(# ({a} \/ the carrier of P),IR,TR #) = {a} \/ the carrier of P & the InternalRel of pcs-Str(# ({a} \/ the carrier of P),IR,TR #) = [:{a}, the carrier of pcs-Str(# ({a} \/ the carrier of P),IR,TR #):] \/ the InternalRel of P & the ToleranceRel of pcs-Str(# ({a} \/ the carrier of P),IR,TR #) = ([:{a}, the carrier of pcs-Str(# ({a} \/ the carrier of P),IR,TR #):] \/ [: the carrier of pcs-Str(# ({a} \/ the carrier of P),IR,TR #),{a}:]) \/ the ToleranceRel of P )
thus  ( the carrier of pcs-Str(# ({a} \/ the carrier of P),IR,TR #) = {a} \/ the carrier of P & the InternalRel of pcs-Str(# ({a} \/ the carrier of P),IR,TR #) = [:{a}, the carrier of pcs-Str(# ({a} \/ the carrier of P),IR,TR #):] \/ the InternalRel of P & the ToleranceRel of pcs-Str(# ({a} \/ the carrier of P),IR,TR #) = ([:{a}, the carrier of pcs-Str(# ({a} \/ the carrier of P),IR,TR #):] \/ [: the carrier of pcs-Str(# ({a} \/ the carrier of P),IR,TR #),{a}:]) \/ the ToleranceRel of P ) ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being strict pcs-Str st the carrier of b1 = {a} \/ the carrier of P & the InternalRel of b1 = [:{a}, the carrier of b1:] \/ the InternalRel of P & the ToleranceRel of b1 = ([:{a}, the carrier of b1:] \/ [: the carrier of b1,{a}:]) \/ the ToleranceRel of P & the carrier of b2 = {a} \/ the carrier of P & the InternalRel of b2 = [:{a}, the carrier of b2:] \/ the InternalRel of P & the ToleranceRel of b2 = ([:{a}, the carrier of b2:] \/ [: the carrier of b2,{a}:]) \/ the ToleranceRel of P holds
b1 = b2 ;
end;

:: deftheorem Def39 defines pcs-extension PCS_0:def_39_:_
 for P being pcs-Str
for a being set
for b3 being strict pcs-Str holds
( b3 = pcs-extension (P,a) iff ( the carrier of b3 = {a} \/ the carrier of P & the InternalRel of b3 = [:{a}, the carrier of b3:] \/ the InternalRel of P & the ToleranceRel of b3 = ([:{a}, the carrier of b3:] \/ [: the carrier of b3,{a}:]) \/ the ToleranceRel of P ) );

registration
let P be pcs-Str ;
let a be set ;
cluster pcs-extension (P,a) ->  non empty strict ;
coherence
 not pcs-extension (P,a) is empty
proof
 the carrier of (pcs-extension (P,a)) = {a} \/ the carrier of P by Def39;
hence  not the carrier of (pcs-extension (P,a)) is empty ; :: according to STRUCT_0:def_1 ::_thesis: verum
end;
end;

theorem Th21: :: PCS_0:21
for P being pcs-Str
for a being set holds
( the carrier of P c= the carrier of (pcs-extension (P,a)) & the InternalRel of P c= the InternalRel of (pcs-extension (P,a)) & the ToleranceRel of P c= the ToleranceRel of (pcs-extension (P,a)) )
proof
let P be pcs-Str ; ::_thesis: for a being set holds
( the carrier of P c= the carrier of (pcs-extension (P,a)) & the InternalRel of P c= the InternalRel of (pcs-extension (P,a)) & the ToleranceRel of P c= the ToleranceRel of (pcs-extension (P,a)) )
let a be set ; ::_thesis: ( the carrier of P c= the carrier of (pcs-extension (P,a)) & the InternalRel of P c= the InternalRel of (pcs-extension (P,a)) & the ToleranceRel of P c= the ToleranceRel of (pcs-extension (P,a)) )
set R = pcs-extension (P,a);
A1: the carrier of (pcs-extension (P,a)) = {a} \/ the carrier of P by Def39;
A2: the InternalRel of (pcs-extension (P,a)) = [:{a}, the carrier of (pcs-extension (P,a)):] \/ the InternalRel of P by Def39;
 the ToleranceRel of (pcs-extension (P,a)) = ([:{a}, the carrier of (pcs-extension (P,a)):] \/ [: the carrier of (pcs-extension (P,a)),{a}:]) \/ the ToleranceRel of P by Def39;
hence  ( the carrier of P c= the carrier of (pcs-extension (P,a)) & the InternalRel of P c= the InternalRel of (pcs-extension (P,a)) & the ToleranceRel of P c= the ToleranceRel of (pcs-extension (P,a)) ) by A1, A2, XBOOLE_1:7; ::_thesis: verum
end;

theorem :: PCS_0:22
for P being pcs-Str
for a being set
for p, q being Element of (pcs-extension (P,a)) st p = a holds
p <= q
proof
let P be pcs-Str ; ::_thesis: for a being set
for p, q being Element of (pcs-extension (P,a)) st p = a holds
p <= q
let a be set ; ::_thesis: for p, q being Element of (pcs-extension (P,a)) st p = a holds
p <= q
set R = pcs-extension (P,a);
let p, q be Element of (pcs-extension (P,a)); ::_thesis: ( p = a implies p <= q )
assume A1: p = a ; ::_thesis: p <= q
A2: the InternalRel of (pcs-extension (P,a)) = [:{a}, the carrier of (pcs-extension (P,a)):] \/ the InternalRel of P by Def39;
 [a,q] in [:{a}, the carrier of (pcs-extension (P,a)):] by ZFMISC_1:105;
hence  [p,q] in the InternalRel of (pcs-extension (P,a)) by A1, A2, XBOOLE_0:def_3; :: according to ORDERS_2:def_5 ::_thesis: verum
end;

theorem Th23: :: PCS_0:23
for P being pcs-Str
for a being set
for p, q being Element of P
for p1, q1 being Element of (pcs-extension (P,a)) st p = p1 & q = q1 & p <= q holds
p1 <= q1
proof
let P be pcs-Str ; ::_thesis: for a being set
for p, q being Element of P
for p1, q1 being Element of (pcs-extension (P,a)) st p = p1 & q = q1 & p <= q holds
p1 <= q1
let a be set ; ::_thesis: for p, q being Element of P
for p1, q1 being Element of (pcs-extension (P,a)) st p = p1 & q = q1 & p <= q holds
p1 <= q1
let p, q be Element of P; ::_thesis: for p1, q1 being Element of (pcs-extension (P,a)) st p = p1 & q = q1 & p <= q holds
p1 <= q1
let p1, q1 be Element of (pcs-extension (P,a)); ::_thesis: ( p = p1 & q = q1 & p <= q implies p1 <= q1 )
assume that
A1: p = p1 and
A2: q = q1 and
A3: [p,q] in the InternalRel of P ; :: according to ORDERS_2:def_5 ::_thesis: p1 <= q1
 the InternalRel of P c= the InternalRel of (pcs-extension (P,a)) by Th21;
hence  [p1,q1] in the InternalRel of (pcs-extension (P,a)) by A1, A2, A3; :: according to ORDERS_2:def_5 ::_thesis: verum
end;

theorem Th24: :: PCS_0:24
for P being pcs-Str
for a being set
for p being Element of P
for p1, q1 being Element of (pcs-extension (P,a)) st p = p1 & p <> a & p1 <= q1 & not a in the carrier of P holds
( q1 in the carrier of P & q1 <> a )
proof
let P be pcs-Str ; ::_thesis: for a being set
for p being Element of P
for p1, q1 being Element of (pcs-extension (P,a)) st p = p1 & p <> a & p1 <= q1 & not a in the carrier of P holds
( q1 in the carrier of P & q1 <> a )
let a be set ; ::_thesis: for p being Element of P
for p1, q1 being Element of (pcs-extension (P,a)) st p = p1 & p <> a & p1 <= q1 & not a in the carrier of P holds
( q1 in the carrier of P & q1 <> a )
let p be Element of P; ::_thesis: for p1, q1 being Element of (pcs-extension (P,a)) st p = p1 & p <> a & p1 <= q1 & not a in the carrier of P holds
( q1 in the carrier of P & q1 <> a )
let p1, q1 be Element of (pcs-extension (P,a)); ::_thesis: ( p = p1 & p <> a & p1 <= q1 & not a in the carrier of P implies ( q1 in the carrier of P & q1 <> a ) )
assume that
A1: p = p1 and
A2: p <> a and
A3: p1 <= q1 and
A4: not a in the carrier of P ; ::_thesis: ( q1 in the carrier of P & q1 <> a )
set R = pcs-extension (P,a);
A5: the InternalRel of (pcs-extension (P,a)) = [:{a}, the carrier of (pcs-extension (P,a)):] \/ the InternalRel of P by Def39;
 [p1,q1] in the InternalRel of (pcs-extension (P,a)) by A3, ORDERS_2:def_5;
then  ( [p1,q1] in [:{a}, the carrier of (pcs-extension (P,a)):] or [p1,q1] in the InternalRel of P ) by A5, XBOOLE_0:def_3;
hence  ( q1 in the carrier of P & q1 <> a ) by A1, A2, A4, ZFMISC_1:87, ZFMISC_1:105; ::_thesis: verum
end;

theorem Th25: :: PCS_0:25
for P being pcs-Str
for a being set
for p being Element of (pcs-extension (P,a)) st p <> a holds
p in the carrier of P
proof
let P be pcs-Str ; ::_thesis: for a being set
for p being Element of (pcs-extension (P,a)) st p <> a holds
p in the carrier of P
let a be set ; ::_thesis: for p being Element of (pcs-extension (P,a)) st p <> a holds
p in the carrier of P
let p be Element of (pcs-extension (P,a)); ::_thesis: ( p <> a implies p in the carrier of P )
assume A1: p <> a ; ::_thesis: p in the carrier of P
 the carrier of (pcs-extension (P,a)) = {a} \/ the carrier of P by Def39;
then  ( p in {a} or p in the carrier of P ) by XBOOLE_0:def_3;
hence  p in the carrier of P by A1, TARSKI:def_1; ::_thesis: verum
end;

theorem Th26: :: PCS_0:26
for P being pcs-Str
for a being set
for p, q being Element of P
for p1, q1 being Element of (pcs-extension (P,a)) st p = p1 & q = q1 & p <> a & p1 <= q1 holds
p <= q
proof
let P be pcs-Str ; ::_thesis: for a being set
for p, q being Element of P
for p1, q1 being Element of (pcs-extension (P,a)) st p = p1 & q = q1 & p <> a & p1 <= q1 holds
p <= q
let a be set ; ::_thesis: for p, q being Element of P
for p1, q1 being Element of (pcs-extension (P,a)) st p = p1 & q = q1 & p <> a & p1 <= q1 holds
p <= q
let p, q be Element of P; ::_thesis: for p1, q1 being Element of (pcs-extension (P,a)) st p = p1 & q = q1 & p <> a & p1 <= q1 holds
p <= q
let p1, q1 be Element of (pcs-extension (P,a)); ::_thesis: ( p = p1 & q = q1 & p <> a & p1 <= q1 implies p <= q )
assume that
A1: p = p1 and
A2: q = q1 and
A3: p <> a and
A4: p1 <= q1 ; ::_thesis: p <= q
set R = pcs-extension (P,a);
A5: the InternalRel of (pcs-extension (P,a)) = [:{a}, the carrier of (pcs-extension (P,a)):] \/ the InternalRel of P by Def39;
 [p1,q1] in the InternalRel of (pcs-extension (P,a)) by A4, ORDERS_2:def_5;
then  ( [p1,q1] in [:{a}, the carrier of (pcs-extension (P,a)):] or [p1,q1] in the InternalRel of P ) by A5, XBOOLE_0:def_3;
hence  [p,q] in the InternalRel of P by A1, A2, A3, ZFMISC_1:105; :: according to ORDERS_2:def_5 ::_thesis: verum
end;

theorem Th27: :: PCS_0:27
for P being pcs-Str
for a being set
for p, q being Element of (pcs-extension (P,a)) st p = a holds
( p (--) q & q (--) p )
proof
let P be pcs-Str ; ::_thesis: for a being set
for p, q being Element of (pcs-extension (P,a)) st p = a holds
( p (--) q & q (--) p )
let a be set ; ::_thesis: for p, q being Element of (pcs-extension (P,a)) st p = a holds
( p (--) q & q (--) p )
set R = pcs-extension (P,a);
let p, q be Element of (pcs-extension (P,a)); ::_thesis: ( p = a implies ( p (--) q & q (--) p ) )
assume A1: p = a ; ::_thesis: ( p (--) q & q (--) p )
 the ToleranceRel of (pcs-extension (P,a)) = ([:{a}, the carrier of (pcs-extension (P,a)):] \/ [: the carrier of (pcs-extension (P,a)),{a}:]) \/ the ToleranceRel of P by Def39;
then A2: the ToleranceRel of (pcs-extension (P,a)) = [:{a}, the carrier of (pcs-extension (P,a)):] \/ ([: the carrier of (pcs-extension (P,a)),{a}:] \/ the ToleranceRel of P) by XBOOLE_1:4;
A3: [a,q] in [:{a}, the carrier of (pcs-extension (P,a)):] by ZFMISC_1:105;
 [q,a] in [: the carrier of (pcs-extension (P,a)),{a}:] by ZFMISC_1:106;
then  [q,a] in [: the carrier of (pcs-extension (P,a)),{a}:] \/ the ToleranceRel of P by XBOOLE_0:def_3;
hence  ( [p,q] in the ToleranceRel of (pcs-extension (P,a)) & [q,p] in the ToleranceRel of (pcs-extension (P,a)) ) by A1, A2, A3, XBOOLE_0:def_3; :: according to PCS_0:def_7 ::_thesis: verum
end;

theorem Th28: :: PCS_0:28
for P being pcs-Str
for a being set
for p, q being Element of P
for p1, q1 being Element of (pcs-extension (P,a)) st p = p1 & q = q1 & p (--) q holds
p1 (--) q1
proof
let P be pcs-Str ; ::_thesis: for a being set
for p, q being Element of P
for p1, q1 being Element of (pcs-extension (P,a)) st p = p1 & q = q1 & p (--) q holds
p1 (--) q1
let a be set ; ::_thesis: for p, q being Element of P
for p1, q1 being Element of (pcs-extension (P,a)) st p = p1 & q = q1 & p (--) q holds
p1 (--) q1
let p, q be Element of P; ::_thesis: for p1, q1 being Element of (pcs-extension (P,a)) st p = p1 & q = q1 & p (--) q holds
p1 (--) q1
let p1, q1 be Element of (pcs-extension (P,a)); ::_thesis: ( p = p1 & q = q1 & p (--) q implies p1 (--) q1 )
assume that
A1: p = p1 and
A2: q = q1 and
A3: [p,q] in the ToleranceRel of P ; :: according to PCS_0:def_7 ::_thesis: p1 (--) q1
 the ToleranceRel of P c= the ToleranceRel of (pcs-extension (P,a)) by Th21;
hence  [p1,q1] in the ToleranceRel of (pcs-extension (P,a)) by A1, A2, A3; :: according to PCS_0:def_7 ::_thesis: verum
end;

theorem Th29: :: PCS_0:29
for P being pcs-Str
for a being set
for p, q being Element of P
for p1, q1 being Element of (pcs-extension (P,a)) st p = p1 & q = q1 & p <> a & q <> a & p1 (--) q1 holds
p (--) q
proof
let P be pcs-Str ; ::_thesis: for a being set
for p, q being Element of P
for p1, q1 being Element of (pcs-extension (P,a)) st p = p1 & q = q1 & p <> a & q <> a & p1 (--) q1 holds
p (--) q
let a be set ; ::_thesis: for p, q being Element of P
for p1, q1 being Element of (pcs-extension (P,a)) st p = p1 & q = q1 & p <> a & q <> a & p1 (--) q1 holds
p (--) q
let p, q be Element of P; ::_thesis: for p1, q1 being Element of (pcs-extension (P,a)) st p = p1 & q = q1 & p <> a & q <> a & p1 (--) q1 holds
p (--) q
let p1, q1 be Element of (pcs-extension (P,a)); ::_thesis: ( p = p1 & q = q1 & p <> a & q <> a & p1 (--) q1 implies p (--) q )
assume that
A1: p = p1 and
A2: q = q1 and
A3: p <> a and
A4: q <> a and
A5: p1 (--) q1 ; ::_thesis: p (--) q
set R = pcs-extension (P,a);
A6: the ToleranceRel of (pcs-extension (P,a)) = ([:{a}, the carrier of (pcs-extension (P,a)):] \/ [: the carrier of (pcs-extension (P,a)),{a}:]) \/ the ToleranceRel of P by Def39;
 [p1,q1] in the ToleranceRel of (pcs-extension (P,a)) by A5, Def7;
then  ( [p1,q1] in [:{a}, the carrier of (pcs-extension (P,a)):] \/ [: the carrier of (pcs-extension (P,a)),{a}:] or [p1,q1] in the ToleranceRel of P ) by A6, XBOOLE_0:def_3;
then  ( [p1,q1] in [:{a}, the carrier of (pcs-extension (P,a)):] or [p1,q1] in [: the carrier of (pcs-extension (P,a)),{a}:] or [p1,q1] in the ToleranceRel of P ) by XBOOLE_0:def_3;
hence  [p,q] in the ToleranceRel of P by A1, A2, A3, A4, ZFMISC_1:105, ZFMISC_1:106; :: according to PCS_0:def_7 ::_thesis: verum
end;

registration
let P be reflexive pcs-Str ;
let a be set ;
cluster pcs-extension (P,a) ->  reflexive strict ;
coherence
 pcs-extension (P,a) is reflexive
proof
set R = pcs-extension (P,a);
A1: the carrier of (pcs-extension (P,a)) = {a} \/ the carrier of P by Def39;
A2: the InternalRel of (pcs-extension (P,a)) = [:{a}, the carrier of (pcs-extension (P,a)):] \/ the InternalRel of P by Def39;
let p be set ; :: according to RELAT_2:def_1,ORDERS_2:def_2 ::_thesis: ( not p in the carrier of (pcs-extension (P,a)) or [^,^] in the InternalRel of (pcs-extension (P,a)) )
assume A3: p in the carrier of (pcs-extension (P,a)) ; ::_thesis: [^,^] in the InternalRel of (pcs-extension (P,a))
percases ( p in {a} or p in the carrier of P ) by A1, A3, XBOOLE_0:def_3;
suppose p in {a} ; ::_thesis: [^,^] in the InternalRel of (pcs-extension (P,a))
then  p = a by TARSKI:def_1;
then  [p,p] in [:{a}, the carrier of (pcs-extension (P,a)):] by A3, ZFMISC_1:105;
hence  [^,^] in the InternalRel of (pcs-extension (P,a)) by A2, XBOOLE_0:def_3; ::_thesis: verum
end;
supposeA4: p in the carrier of P ; ::_thesis: [^,^] in the InternalRel of (pcs-extension (P,a))
 the InternalRel of P is_reflexive_in the carrier of P by ORDERS_2:def_2;
then  [p,p] in the InternalRel of P by A4, RELAT_2:def_1;
hence  [^,^] in the InternalRel of (pcs-extension (P,a)) by A2, XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
end;

theorem Th30: :: PCS_0:30
for P being transitive pcs-Str
for a being set st not a in the carrier of P holds
pcs-extension (P,a) is transitive
proof
let P be transitive pcs-Str ; ::_thesis: for a being set st not a in the carrier of P holds
pcs-extension (P,a) is transitive
let a be set ; ::_thesis: ( not a in the carrier of P implies pcs-extension (P,a) is transitive )
assume A1: not a in the carrier of P ; ::_thesis: pcs-extension (P,a) is transitive
set R = pcs-extension (P,a);
A2: the InternalRel of (pcs-extension (P,a)) = [:{a}, the carrier of (pcs-extension (P,a)):] \/ the InternalRel of P by Def39;
let x, y, z be set ; :: according to RELAT_2:def_8,ORDERS_2:def_3 ::_thesis: ( not x in the carrier of (pcs-extension (P,a)) or not y in the carrier of (pcs-extension (P,a)) or not z in the carrier of (pcs-extension (P,a)) or not [^,^] in the InternalRel of (pcs-extension (P,a)) or not [^,^] in the InternalRel of (pcs-extension (P,a)) or [^,^] in the InternalRel of (pcs-extension (P,a)) )
assume that
A3: x in the carrier of (pcs-extension (P,a)) and
A4: y in the carrier of (pcs-extension (P,a)) and
A5: z in the carrier of (pcs-extension (P,a)) and
A6: [x,y] in the InternalRel of (pcs-extension (P,a)) and
A7: [y,z] in the InternalRel of (pcs-extension (P,a)) ; ::_thesis: [^,^] in the InternalRel of (pcs-extension (P,a))
A8: [a,z] in [:{a}, the carrier of (pcs-extension (P,a)):] by A5, ZFMISC_1:105;
reconsider x = x, y = y, z = z as Element of (pcs-extension (P,a)) by A3, A4, A5;
A9: x <= y by A6, ORDERS_2:def_5;
A10: y <= z by A7, ORDERS_2:def_5;
percases ( x = a or x <> a ) ;
suppose x = a ; ::_thesis: [^,^] in the InternalRel of (pcs-extension (P,a))
hence  [^,^] in the InternalRel of (pcs-extension (P,a)) by A2, A8, XBOOLE_0:def_3; ::_thesis: verum
end;
supposeA11: x <> a ; ::_thesis: [^,^] in the InternalRel of (pcs-extension (P,a))
then reconsider x0 = x as Element of P by Th25;
A12: x0 <> a by A11;
then reconsider y0 = y as Element of P by A1, A9, Th24;
 y0 <> a by A1, A9, A12, Th24;
then reconsider z0 = z as Element of P by A1, A10, Th24;
A13: y <> a by A1, A9, A12, Th24;
A14: x0 <= y0 by A9, A11, Th26;
 y0 <= z0 by A10, A13, Th26;
then  x0 <= z0 by A14, YELLOW_0:def_2;
then  x <= z by Th23;
hence  [^,^] in the InternalRel of (pcs-extension (P,a)) by ORDERS_2:def_5; ::_thesis: verum
end;
end;
end;

registration
let P be pcs-tol-reflexive pcs-Str ;
let a be set ;
cluster pcs-extension (P,a) ->  pcs-tol-reflexive strict ;
coherence
 pcs-extension (P,a) is pcs-tol-reflexive
proof
set R = pcs-extension (P,a);
A1: the carrier of (pcs-extension (P,a)) = {a} \/ the carrier of P by Def39;
A2: the ToleranceRel of (pcs-extension (P,a)) = ([:{a}, the carrier of (pcs-extension (P,a)):] \/ [: the carrier of (pcs-extension (P,a)),{a}:]) \/ the ToleranceRel of P by Def39;
then A3: the ToleranceRel of (pcs-extension (P,a)) = [:{a}, the carrier of (pcs-extension (P,a)):] \/ ([: the carrier of (pcs-extension (P,a)),{a}:] \/ the ToleranceRel of P) by XBOOLE_1:4;
let p be set ; :: according to RELAT_2:def_1,PCS_0:def_9 ::_thesis: ( not p in the carrier of (pcs-extension (P,a)) or [^,^] in the ToleranceRel of (pcs-extension (P,a)) )
assume A4: p in the carrier of (pcs-extension (P,a)) ; ::_thesis: [^,^] in the ToleranceRel of (pcs-extension (P,a))
percases ( p in {a} or p in the carrier of P ) by A1, A4, XBOOLE_0:def_3;
suppose p in {a} ; ::_thesis: [^,^] in the ToleranceRel of (pcs-extension (P,a))
then  p = a by TARSKI:def_1;
then  [p,p] in [:{a}, the carrier of (pcs-extension (P,a)):] by A4, ZFMISC_1:105;
hence  [^,^] in the ToleranceRel of (pcs-extension (P,a)) by A3, XBOOLE_0:def_3; ::_thesis: verum
end;
supposeA5: p in the carrier of P ; ::_thesis: [^,^] in the ToleranceRel of (pcs-extension (P,a))
 the ToleranceRel of P is_reflexive_in the carrier of P by Def9;
then  [p,p] in the ToleranceRel of P by A5, RELAT_2:def_1;
hence  [^,^] in the ToleranceRel of (pcs-extension (P,a)) by A2, XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
end;

registration
let P be pcs-tol-symmetric pcs-Str ;
let a be set ;
cluster pcs-extension (P,a) ->  pcs-tol-symmetric strict ;
coherence
 pcs-extension (P,a) is pcs-tol-symmetric
proof
set R = pcs-extension (P,a);
A1: the ToleranceRel of (pcs-extension (P,a)) = ([:{a}, the carrier of (pcs-extension (P,a)):] \/ [: the carrier of (pcs-extension (P,a)),{a}:]) \/ the ToleranceRel of P by Def39;
let p, q be set ; :: according to RELAT_2:def_3,PCS_0:def_11 ::_thesis: ( not p in the carrier of (pcs-extension (P,a)) or not q in the carrier of (pcs-extension (P,a)) or not [^,^] in the ToleranceRel of (pcs-extension (P,a)) or [^,^] in the ToleranceRel of (pcs-extension (P,a)) )
assume that
 p in the carrier of (pcs-extension (P,a)) and
 q in the carrier of (pcs-extension (P,a)) and
A2: [p,q] in the ToleranceRel of (pcs-extension (P,a)) ; ::_thesis: [^,^] in the ToleranceRel of (pcs-extension (P,a))
A3: the ToleranceRel of P is_symmetric_in the carrier of P by Def11;
percases ( [p,q] in [:{a}, the carrier of (pcs-extension (P,a)):] \/ [: the carrier of (pcs-extension (P,a)),{a}:] or [p,q] in the ToleranceRel of P ) by A1, A2, XBOOLE_0:def_3;
supposeA4: [p,q] in [:{a}, the carrier of (pcs-extension (P,a)):] \/ [: the carrier of (pcs-extension (P,a)),{a}:] ; ::_thesis: [^,^] in the ToleranceRel of (pcs-extension (P,a))
percases ( [p,q] in [:{a}, the carrier of (pcs-extension (P,a)):] or [p,q] in [: the carrier of (pcs-extension (P,a)),{a}:] ) by A4, XBOOLE_0:def_3;
supposeA5: [p,q] in [:{a}, the carrier of (pcs-extension (P,a)):] ; ::_thesis: [^,^] in the ToleranceRel of (pcs-extension (P,a))
then A6: p = a by ZFMISC_1:105;
 q in the carrier of (pcs-extension (P,a)) by A5, ZFMISC_1:105;
then  [q,p] in [: the carrier of (pcs-extension (P,a)),{a}:] by A6, ZFMISC_1:106;
then  [q,p] in [:{a}, the carrier of (pcs-extension (P,a)):] \/ [: the carrier of (pcs-extension (P,a)),{a}:] by XBOOLE_0:def_3;
hence  [^,^] in the ToleranceRel of (pcs-extension (P,a)) by A1, XBOOLE_0:def_3; ::_thesis: verum
end;
supposeA7: [p,q] in [: the carrier of (pcs-extension (P,a)),{a}:] ; ::_thesis: [^,^] in the ToleranceRel of (pcs-extension (P,a))
then A8: q = a by ZFMISC_1:106;
 p in the carrier of (pcs-extension (P,a)) by A7, ZFMISC_1:106;
then  [q,p] in [:{a}, the carrier of (pcs-extension (P,a)):] by A8, ZFMISC_1:105;
then  [q,p] in [:{a}, the carrier of (pcs-extension (P,a)):] \/ [: the carrier of (pcs-extension (P,a)),{a}:] by XBOOLE_0:def_3;
hence  [^,^] in the ToleranceRel of (pcs-extension (P,a)) by A1, XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
supposeA9: [p,q] in the ToleranceRel of P ; ::_thesis: [^,^] in the ToleranceRel of (pcs-extension (P,a))
then A10: p in the carrier of P by ZFMISC_1:87;
 q in the carrier of P by A9, ZFMISC_1:87;
then  [q,p] in the ToleranceRel of P by A3, A9, A10, RELAT_2:def_3;
hence  [^,^] in the ToleranceRel of (pcs-extension (P,a)) by A1, XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
end;

theorem Th31: :: PCS_0:31
for P being pcs-compatible pcs-Str
for a being set st not a in the carrier of P holds
pcs-extension (P,a) is pcs-compatible
proof
let P be pcs-compatible pcs-Str ; ::_thesis: for a being set st not a in the carrier of P holds
pcs-extension (P,a) is pcs-compatible
let a be set ; ::_thesis: ( not a in the carrier of P implies pcs-extension (P,a) is pcs-compatible )
assume A1: not a in the carrier of P ; ::_thesis: pcs-extension (P,a) is pcs-compatible
set R = pcs-extension (P,a);
let p, p9, q, q9 be Element of (pcs-extension (P,a)); :: according to PCS_0:def_22 ::_thesis: ( p (--) q & p9 <= p & q9 <= q implies p9 (--) q9 )
assume that
A2: p (--) q and
A3: p9 <= p and
A4: q9 <= q ; ::_thesis: p9 (--) q9
percases ( p9 = a or q9 = a or ( p9 <> a & q9 <> a ) ) ;
suppose ( p9 = a or q9 = a ) ; ::_thesis: p9 (--) q9
hence  p9 (--) q9 by Th27; ::_thesis: verum
end;
supposethat A5: p9 <> a and
A6: q9 <> a ; ::_thesis: p9 (--) q9
reconsider p90 = p9, q90 = q9 as Element of P by A5, A6, Th25;
A7: p90 <> a by A5;
A8: q90 <> a by A6;
A9: p <> a by A1, A3, A7, Th24;
A10: q <> a by A1, A4, A8, Th24;
reconsider p0 = p, q0 = q as Element of P by A1, A3, A4, A7, A8, Th24;
A11: p0 (--) q0 by A2, A9, A10, Th29;
A12: p90 <= p0 by A3, A5, Th26;
 q90 <= q0 by A4, A6, Th26;
then  p90 (--) q90 by A11, A12, Def22;
hence  p9 (--) q9 by Th28; ::_thesis: verum
end;
end;
end;

theorem :: PCS_0:32
for P being pcs
for a being set st not a in the carrier of P holds
pcs-extension (P,a) is pcs
proof
let P be pcs; ::_thesis: for a being set st not a in the carrier of P holds
pcs-extension (P,a) is pcs
let a be set ; ::_thesis: ( not a in the carrier of P implies pcs-extension (P,a) is pcs )
assume A1: not a in the carrier of P ; ::_thesis: pcs-extension (P,a) is pcs
set R = pcs-extension (P,a);
 ( pcs-extension (P,a) is reflexive & pcs-extension (P,a) is transitive & pcs-extension (P,a) is pcs-tol-reflexive & pcs-extension (P,a) is pcs-tol-symmetric & pcs-extension (P,a) is pcs-compatible ) by A1, Th30, Th31;
hence  pcs-extension (P,a) is pcs ; ::_thesis: verum
end;

definition
let P be pcs-Str ;
func pcs-reverse P ->  strict pcs-Str  means :Def40: :: PCS_0:def 40
( the carrier of it = the carrier of P & the InternalRel of it = the InternalRel of P ~ & the ToleranceRel of it = the ToleranceRel of P ` );
existence
 ex b1 being strict pcs-Str st
( the carrier of b1 = the carrier of P & the InternalRel of b1 = the InternalRel of P ~ & the ToleranceRel of b1 = the ToleranceRel of P ` )
proof
reconsider TR = the ToleranceRel of P ` as Relation of the carrier of P ;
take  pcs-Str(# the carrier of P,( the InternalRel of P ~),TR #) ; ::_thesis: ( the carrier of pcs-Str(# the carrier of P,( the InternalRel of P ~),TR #) = the carrier of P & the InternalRel of pcs-Str(# the carrier of P,( the InternalRel of P ~),TR #) = the InternalRel of P ~ & the ToleranceRel of pcs-Str(# the carrier of P,( the InternalRel of P ~),TR #) = the ToleranceRel of P ` )
thus  ( the carrier of pcs-Str(# the carrier of P,( the InternalRel of P ~),TR #) = the carrier of P & the InternalRel of pcs-Str(# the carrier of P,( the InternalRel of P ~),TR #) = the InternalRel of P ~ & the ToleranceRel of pcs-Str(# the carrier of P,( the InternalRel of P ~),TR #) = the ToleranceRel of P ` ) ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being strict pcs-Str st the carrier of b1 = the carrier of P & the InternalRel of b1 = the InternalRel of P ~ & the ToleranceRel of b1 = the ToleranceRel of P ` & the carrier of b2 = the carrier of P & the InternalRel of b2 = the InternalRel of P ~ & the ToleranceRel of b2 = the ToleranceRel of P ` holds
b1 = b2 ;
end;

:: deftheorem Def40 defines pcs-reverse PCS_0:def_40_:_
 for P being pcs-Str
for b2 being strict pcs-Str holds
( b2 = pcs-reverse P iff ( the carrier of b2 = the carrier of P & the InternalRel of b2 = the InternalRel of P ~ & the ToleranceRel of b2 = the ToleranceRel of P ` ) );

registration
let P be non empty pcs-Str ;
cluster pcs-reverse P ->  non empty strict ;
coherence
 not pcs-reverse P is empty
proof
 the carrier of (pcs-reverse P) = the carrier of P by Def40;
hence  not the carrier of (pcs-reverse P) is empty ; :: according to STRUCT_0:def_1 ::_thesis: verum
end;
end;

theorem Th33: :: PCS_0:33
for P being pcs-Str
for p, q being Element of P
for p9, q9 being Element of (pcs-reverse P) st p = p9 & q = q9 holds
( p <= q iff q9 <= p9 )
proof
let P be pcs-Str ; ::_thesis: for p, q being Element of P
for p9, q9 being Element of (pcs-reverse P) st p = p9 & q = q9 holds
( p <= q iff q9 <= p9 )
let p, q be Element of P; ::_thesis: for p9, q9 being Element of (pcs-reverse P) st p = p9 & q = q9 holds
( p <= q iff q9 <= p9 )
set R = pcs-reverse P;
let p9, q9 be Element of (pcs-reverse P); ::_thesis: ( p = p9 & q = q9 implies ( p <= q iff q9 <= p9 ) )
assume that
A1: p = p9 and
A2: q = q9 ; ::_thesis: ( p <= q iff q9 <= p9 )
A3: the InternalRel of (pcs-reverse P) = the InternalRel of P ~ by Def40;
thus  ( p <= q implies q9 <= p9 ) ::_thesis: ( q9 <= p9 implies p <= q )
proof
assume  [p,q] in the InternalRel of P ; :: according to ORDERS_2:def_5 ::_thesis: q9 <= p9
hence  [q9,p9] in the InternalRel of (pcs-reverse P) by A1, A2, A3, RELAT_1:def_7; :: according to ORDERS_2:def_5 ::_thesis: verum
end;
assume  [q9,p9] in the InternalRel of (pcs-reverse P) ; :: according to ORDERS_2:def_5 ::_thesis: p <= q
hence  [p,q] in the InternalRel of P by A1, A2, A3, RELAT_1:def_7; :: according to ORDERS_2:def_5 ::_thesis: verum
end;

theorem Th34: :: PCS_0:34
for P being pcs-Str
for p, q being Element of P
for p9, q9 being Element of (pcs-reverse P) st p = p9 & q = q9 & p (--) q holds
not p9 (--) q9
proof
let P be pcs-Str ; ::_thesis: for p, q being Element of P
for p9, q9 being Element of (pcs-reverse P) st p = p9 & q = q9 & p (--) q holds
not p9 (--) q9
let p, q be Element of P; ::_thesis: for p9, q9 being Element of (pcs-reverse P) st p = p9 & q = q9 & p (--) q holds
not p9 (--) q9
set R = pcs-reverse P;
let p9, q9 be Element of (pcs-reverse P); ::_thesis: ( p = p9 & q = q9 & p (--) q implies not p9 (--) q9 )
assume that
A1: p = p9 and
A2: q = q9 ; ::_thesis: ( not p (--) q or not p9 (--) q9 )
A3: the ToleranceRel of (pcs-reverse P) = the ToleranceRel of P ` by Def40;
assume  [p,q] in the ToleranceRel of P ; :: according to PCS_0:def_7 ::_thesis: not p9 (--) q9
hence  not [p9,q9] in the ToleranceRel of (pcs-reverse P) by A1, A2, A3, XBOOLE_0:def_5; :: according to PCS_0:def_7 ::_thesis: verum
end;

theorem Th35: :: PCS_0:35
for P being non empty pcs-Str
for p, q being Element of P
for p9, q9 being Element of (pcs-reverse P) st p = p9 & q = q9 & not p9 (--) q9 holds
p (--) q
proof
let P be non empty pcs-Str ; ::_thesis: for p, q being Element of P
for p9, q9 being Element of (pcs-reverse P) st p = p9 & q = q9 & not p9 (--) q9 holds
p (--) q
let p, q be Element of P; ::_thesis: for p9, q9 being Element of (pcs-reverse P) st p = p9 & q = q9 & not p9 (--) q9 holds
p (--) q
set R = pcs-reverse P;
let p9, q9 be Element of (pcs-reverse P); ::_thesis: ( p = p9 & q = q9 & not p9 (--) q9 implies p (--) q )
assume that
A1: p = p9 and
A2: q = q9 ; ::_thesis: ( p9 (--) q9 or p (--) q )
A3: the ToleranceRel of (pcs-reverse P) = the ToleranceRel of P ` by Def40;
assume A4: not [p9,q9] in the ToleranceRel of (pcs-reverse P) ; :: according to PCS_0:def_7 ::_thesis: p (--) q
 [p,q] in [: the carrier of P, the carrier of P:] by ZFMISC_1:87;
hence  [p,q] in the ToleranceRel of P by A1, A2, A3, A4, XBOOLE_0:def_5; :: according to PCS_0:def_7 ::_thesis: verum
end;

registration
let P be reflexive pcs-Str ;
cluster pcs-reverse P ->  reflexive strict ;
coherence
 pcs-reverse P is reflexive
proof
set R = pcs-reverse P;
A1: the carrier of (pcs-reverse P) = the carrier of P by Def40;
A2: the InternalRel of (pcs-reverse P) = the InternalRel of P ~ by Def40;
 the InternalRel of P is_reflexive_in the carrier of P by ORDERS_2:def_2;
hence  the InternalRel of (pcs-reverse P) is_reflexive_in the carrier of (pcs-reverse P) by A1, A2, ORDERS_1:79; :: according to ORDERS_2:def_2 ::_thesis: verum
end;
end;

registration
let P be transitive pcs-Str ;
cluster pcs-reverse P ->  transitive strict ;
coherence
 pcs-reverse P is transitive
proof
set R = pcs-reverse P;
A1: the carrier of (pcs-reverse P) = the carrier of P by Def40;
A2: the InternalRel of (pcs-reverse P) = the InternalRel of P ~ by Def40;
 the InternalRel of P is_transitive_in the carrier of P by ORDERS_2:def_3;
hence  the InternalRel of (pcs-reverse P) is_transitive_in the carrier of (pcs-reverse P) by A1, A2, ORDERS_1:80; :: according to ORDERS_2:def_3 ::_thesis: verum
end;
end;

registration
let P be pcs-tol-reflexive pcs-Str ;
cluster pcs-reverse P ->  pcs-tol-irreflexive strict ;
coherence
 pcs-reverse P is pcs-tol-irreflexive
proof
set R = pcs-reverse P;
A1: the carrier of (pcs-reverse P) = the carrier of P by Def40;
A2: the ToleranceRel of (pcs-reverse P) = the ToleranceRel of P ` by Def40;
A3: the ToleranceRel of P is_reflexive_in the carrier of P by Def9;
let x be set ; :: according to RELAT_2:def_2,PCS_0:def_10 ::_thesis: ( not x in the carrier of (pcs-reverse P) or not [^,^] in the ToleranceRel of (pcs-reverse P) )
assume  x in the carrier of (pcs-reverse P) ; ::_thesis: not [^,^] in the ToleranceRel of (pcs-reverse P)
then  [x,x] in the ToleranceRel of P by A1, A3, RELAT_2:def_1;
hence  not [^,^] in the ToleranceRel of (pcs-reverse P) by A2, XBOOLE_0:def_5; ::_thesis: verum
end;
end;

registration
let P be pcs-tol-irreflexive pcs-Str ;
cluster pcs-reverse P ->  pcs-tol-reflexive strict ;
coherence
 pcs-reverse P is pcs-tol-reflexive
proof
set R = pcs-reverse P;
A1: the carrier of (pcs-reverse P) = the carrier of P by Def40;
A2: the ToleranceRel of (pcs-reverse P) = the ToleranceRel of P ` by Def40;
A3: the ToleranceRel of P is_irreflexive_in the carrier of P by Def10;
let x be set ; :: according to RELAT_2:def_1,PCS_0:def_9 ::_thesis: ( not x in the carrier of (pcs-reverse P) or [^,^] in the ToleranceRel of (pcs-reverse P) )
assume A4: x in the carrier of (pcs-reverse P) ; ::_thesis: [^,^] in the ToleranceRel of (pcs-reverse P)
then A5: not [x,x] in the ToleranceRel of P by A1, A3, RELAT_2:def_2;
 [x,x] in [: the carrier of (pcs-reverse P), the carrier of (pcs-reverse P):] by A4, ZFMISC_1:87;
hence  [^,^] in the ToleranceRel of (pcs-reverse P) by A1, A2, A5, XBOOLE_0:def_5; ::_thesis: verum
end;
end;

registration
let P be pcs-tol-symmetric pcs-Str ;
cluster pcs-reverse P ->  pcs-tol-symmetric strict ;
coherence
 pcs-reverse P is pcs-tol-symmetric
proof
set R = pcs-reverse P;
A1: the carrier of (pcs-reverse P) = the carrier of P by Def40;
A2: the ToleranceRel of (pcs-reverse P) = the ToleranceRel of P ` by Def40;
A3: the ToleranceRel of P is_symmetric_in the carrier of P by Def11;
let x, y be set ; :: according to RELAT_2:def_3,PCS_0:def_11 ::_thesis: ( not x in the carrier of (pcs-reverse P) or not y in the carrier of (pcs-reverse P) or not [^,^] in the ToleranceRel of (pcs-reverse P) or [^,^] in the ToleranceRel of (pcs-reverse P) )
assume that
A4: x in the carrier of (pcs-reverse P) and
A5: y in the carrier of (pcs-reverse P) ; ::_thesis: ( not [^,^] in the ToleranceRel of (pcs-reverse P) or [^,^] in the ToleranceRel of (pcs-reverse P) )
assume  [x,y] in the ToleranceRel of (pcs-reverse P) ; ::_thesis: [^,^] in the ToleranceRel of (pcs-reverse P)
then  not [x,y] in the ToleranceRel of P by A2, XBOOLE_0:def_5;
then A6: not [y,x] in the ToleranceRel of P by A1, A3, A4, A5, RELAT_2:def_3;
 [y,x] in [: the carrier of P, the carrier of P:] by A1, A4, A5, ZFMISC_1:87;
hence  [^,^] in the ToleranceRel of (pcs-reverse P) by A2, A6, XBOOLE_0:def_5; ::_thesis: verum
end;
end;

registration
let P be pcs-compatible pcs-Str ;
cluster pcs-reverse P ->  strict pcs-compatible ;
coherence
 pcs-reverse P is pcs-compatible
proof
set R = pcs-reverse P;
A1: the carrier of (pcs-reverse P) = the carrier of P by Def40;
A2: the InternalRel of (pcs-reverse P) = the InternalRel of P ~ by Def40;
A3: the ToleranceRel of (pcs-reverse P) = the ToleranceRel of P ` by Def40;
let p, p9, q, q9 be Element of (pcs-reverse P); :: according to PCS_0:def_22 ::_thesis: ( p (--) q & p9 <= p & q9 <= q implies p9 (--) q9 )
assume that
A4: [p,q] in the ToleranceRel of (pcs-reverse P) and
A5: [p9,p] in the InternalRel of (pcs-reverse P) and
A6: [q9,q] in the InternalRel of (pcs-reverse P) ; :: according to ORDERS_2:def_5,PCS_0:def_7 ::_thesis: p9 (--) q9
A7: p9 in the carrier of (pcs-reverse P) by A5, ZFMISC_1:87;
reconsider p90 = p9, q90 = q9, p0 = p, q0 = q as Element of P by Def40;
 not [p0,q0] in the ToleranceRel of P by A3, A4, XBOOLE_0:def_5;
then A8: not p0 (--) q0 by Def7;
A9: [p0,p90] in the InternalRel of P by A2, A5, RELAT_1:def_7;
A10: [q0,q90] in the InternalRel of P by A2, A6, RELAT_1:def_7;
A11: p0 <= p90 by A9, ORDERS_2:def_5;
 q0 <= q90 by A10, ORDERS_2:def_5;
then  not p90 (--) q90 by A8, A11, Def22;
then A12: not [p90,q90] in the ToleranceRel of P by Def7;
 [p9,q9] in [: the carrier of P, the carrier of P:] by A1, A7, ZFMISC_1:87;
hence  [p9,q9] in the ToleranceRel of (pcs-reverse P) by A3, A12, XBOOLE_0:def_5; :: according to PCS_0:def_7 ::_thesis: verum
end;
end;

definition
let P, Q be pcs-Str ;
funcP pcs-times Q ->  pcs-Str  equals :: PCS_0:def 41
 pcs-Str(# [: the carrier of P, the carrier of Q:],[" the InternalRel of P, the InternalRel of Q"],[^ the ToleranceRel of P, the ToleranceRel of Q^] #);
coherence
 pcs-Str(# [: the carrier of P, the carrier of Q:],[" the InternalRel of P, the InternalRel of Q"],[^ the ToleranceRel of P, the ToleranceRel of Q^] #) is pcs-Str ;
end;

:: deftheorem  defines pcs-times PCS_0:def_41_:_
 for P, Q being pcs-Str holds P pcs-times Q = pcs-Str(# [: the carrier of P, the carrier of Q:],[" the InternalRel of P, the InternalRel of Q"],[^ the ToleranceRel of P, the ToleranceRel of Q^] #);

registration
let P, Q be pcs-Str ;
clusterP pcs-times Q ->  strict ;
coherence
 P pcs-times Q is strict ;
end;

registration
let P, Q be non empty pcs-Str ;
clusterP pcs-times Q ->  non empty ;
coherence
 not P pcs-times Q is empty ;
end;

theorem :: PCS_0:36
for P, Q being pcs-Str
for p, q being Element of (P pcs-times Q)
for p1, p2 being Element of P
for q1, q2 being Element of Q st p = [p1,q1] & q = [p2,q2] holds
( p <= q iff ( p1 <= p2 & q1 <= q2 ) )
proof
let P, Q be pcs-Str ; ::_thesis: for p, q being Element of (P pcs-times Q)
for p1, p2 being Element of P
for q1, q2 being Element of Q st p = [p1,q1] & q = [p2,q2] holds
( p <= q iff ( p1 <= p2 & q1 <= q2 ) )
let p, q be Element of (P pcs-times Q); ::_thesis: for p1, p2 being Element of P
for q1, q2 being Element of Q st p = [p1,q1] & q = [p2,q2] holds
( p <= q iff ( p1 <= p2 & q1 <= q2 ) )
let p1, p2 be Element of P; ::_thesis: for q1, q2 being Element of Q st p = [p1,q1] & q = [p2,q2] holds
( p <= q iff ( p1 <= p2 & q1 <= q2 ) )
let q1, q2 be Element of Q; ::_thesis: ( p = [p1,q1] & q = [p2,q2] implies ( p <= q iff ( p1 <= p2 & q1 <= q2 ) ) )
assume that
A1: p = [p1,q1] and
A2: q = [p2,q2] ; ::_thesis: ( p <= q iff ( p1 <= p2 & q1 <= q2 ) )
thus  ( p <= q implies ( p1 <= p2 & q1 <= q2 ) ) ::_thesis: ( p1 <= p2 & q1 <= q2 implies p <= q )
proof
assume  p <= q ; ::_thesis: ( p1 <= p2 & q1 <= q2 )
then  [p,q] in [" the InternalRel of P, the InternalRel of Q"] by ORDERS_2:def_5;
then consider a, b, s, t being set  such that
A3: p = [a,b] and
A4: q = [s,t] and
A5: [a,s] in the InternalRel of P and
A6: [b,t] in the InternalRel of Q by YELLOW_3:def_1;
A7: a = p1 by A1, A3, XTUPLE_0:1;
A8: b = q1 by A1, A3, XTUPLE_0:1;
thus  [p1,p2] in the InternalRel of P by A2, A4, A5, A7, XTUPLE_0:1; :: according to ORDERS_2:def_5 ::_thesis: q1 <= q2
thus  [q1,q2] in the InternalRel of Q by A2, A4, A6, A8, XTUPLE_0:1; :: according to ORDERS_2:def_5 ::_thesis: verum
end;
assume that
A9: p1 <= p2 and
A10: q1 <= q2 ; ::_thesis: p <= q
A11: [p1,p2] in the InternalRel of P by A9, ORDERS_2:def_5;
 [q1,q2] in the InternalRel of Q by A10, ORDERS_2:def_5;
hence  [p,q] in the InternalRel of (P pcs-times Q) by A1, A2, A11, YELLOW_3:def_1; :: according to ORDERS_2:def_5 ::_thesis: verum
end;

theorem :: PCS_0:37
for P, Q being pcs-Str
for p, q being Element of (P pcs-times Q)
for p1, p2 being Element of P
for q1, q2 being Element of Q st p = [p1,q1] & q = [p2,q2] & p (--) q & not p1 (--) p2 holds
q1 (--) q2
proof
let P, Q be pcs-Str ; ::_thesis: for p, q being Element of (P pcs-times Q)
for p1, p2 being Element of P
for q1, q2 being Element of Q st p = [p1,q1] & q = [p2,q2] & p (--) q & not p1 (--) p2 holds
q1 (--) q2
let p, q be Element of (P pcs-times Q); ::_thesis: for p1, p2 being Element of P
for q1, q2 being Element of Q st p = [p1,q1] & q = [p2,q2] & p (--) q & not p1 (--) p2 holds
q1 (--) q2
let p1, p2 be Element of P; ::_thesis: for q1, q2 being Element of Q st p = [p1,q1] & q = [p2,q2] & p (--) q & not p1 (--) p2 holds
q1 (--) q2
let q1, q2 be Element of Q; ::_thesis: ( p = [p1,q1] & q = [p2,q2] & p (--) q & not p1 (--) p2 implies q1 (--) q2 )
assume that
A1: p = [p1,q1] and
A2: q = [p2,q2] ; ::_thesis: ( not p (--) q or p1 (--) p2 or q1 (--) q2 )
assume  p (--) q ; ::_thesis: ( p1 (--) p2 or q1 (--) q2 )
then  [p,q] in [^ the ToleranceRel of P, the ToleranceRel of Q^] by Def7;
then consider a, b, c, d being set  such that
A3: p = [a,b] and
A4: q = [c,d] and
 a in the carrier of P and
 b in the carrier of Q and
 c in the carrier of P and
 d in the carrier of Q and
A5: ( [a,c] in the ToleranceRel of P or [b,d] in the ToleranceRel of Q ) by Def2;
A6: a = p1 by A1, A3, XTUPLE_0:1;
A7: b = q1 by A1, A3, XTUPLE_0:1;
A8: c = p2 by A2, A4, XTUPLE_0:1;
 d = q2 by A2, A4, XTUPLE_0:1;
hence  ( p1 (--) p2 or q1 (--) q2 ) by A5, A6, A7, A8, Def7; ::_thesis: verum
end;

theorem Th38: :: PCS_0:38
for P, Q being non empty pcs-Str
for p, q being Element of (P pcs-times Q)
for p1, p2 being Element of P
for q1, q2 being Element of Q st p = [p1,q1] & q = [p2,q2] & ( p1 (--) p2 or q1 (--) q2 ) holds
p (--) q
proof
let P, Q be non empty pcs-Str ; ::_thesis: for p, q being Element of (P pcs-times Q)
for p1, p2 being Element of P
for q1, q2 being Element of Q st p = [p1,q1] & q = [p2,q2] & ( p1 (--) p2 or q1 (--) q2 ) holds
p (--) q
let p, q be Element of (P pcs-times Q); ::_thesis: for p1, p2 being Element of P
for q1, q2 being Element of Q st p = [p1,q1] & q = [p2,q2] & ( p1 (--) p2 or q1 (--) q2 ) holds
p (--) q
let p1, p2 be Element of P; ::_thesis: for q1, q2 being Element of Q st p = [p1,q1] & q = [p2,q2] & ( p1 (--) p2 or q1 (--) q2 ) holds
p (--) q
let q1, q2 be Element of Q; ::_thesis: ( p = [p1,q1] & q = [p2,q2] & ( p1 (--) p2 or q1 (--) q2 ) implies p (--) q )
assume that
A1: p = [p1,q1] and
A2: q = [p2,q2] ; ::_thesis: ( ( not p1 (--) p2 & not q1 (--) q2 ) or p (--) q )
assume  ( p1 (--) p2 or q1 (--) q2 ) ; ::_thesis: p (--) q
then  ( [p1,p2] in the ToleranceRel of P or [q1,q2] in the ToleranceRel of Q ) by Def7;
hence  [p,q] in the ToleranceRel of (P pcs-times Q) by A1, A2, Def2; :: according to PCS_0:def_7 ::_thesis: verum
end;

registration
let P, Q be reflexive pcs-Str ;
clusterP pcs-times Q ->  reflexive ;
coherence
 P pcs-times Q is reflexive
proof
 RelStr(# the carrier of (P pcs-times Q), the InternalRel of (P pcs-times Q) #) = [:P,Q:] by YELLOW_3:def_2;
hence  P pcs-times Q is reflexive by WAYBEL_8:12; ::_thesis: verum
end;
end;

registration
let P, Q be transitive pcs-Str ;
clusterP pcs-times Q ->  transitive ;
coherence
 P pcs-times Q is transitive
proof
 RelStr(# the carrier of (P pcs-times Q), the InternalRel of (P pcs-times Q) #) = [:P,Q:] by YELLOW_3:def_2;
hence  P pcs-times Q is transitive by WAYBEL_8:13; ::_thesis: verum
end;
end;

registration
let P be pcs-Str ;
let Q be pcs-tol-reflexive pcs-Str ;
clusterP pcs-times Q ->  pcs-tol-reflexive ;
coherence
 P pcs-times Q is pcs-tol-reflexive
proof
 TolStr(# the carrier of (P pcs-times Q), the ToleranceRel of (P pcs-times Q) #) = [^P,Q^] ;
hence  P pcs-times Q is pcs-tol-reflexive by Th3; ::_thesis: verum
end;
end;

registration
let P be pcs-tol-reflexive pcs-Str ;
let Q be pcs-Str ;
clusterP pcs-times Q ->  pcs-tol-reflexive ;
coherence
 P pcs-times Q is pcs-tol-reflexive
proof
 TolStr(# the carrier of (P pcs-times Q), the ToleranceRel of (P pcs-times Q) #) = [^P,Q^] ;
hence  P pcs-times Q is pcs-tol-reflexive by Th3; ::_thesis: verum
end;
end;

registration
let P, Q be pcs-tol-symmetric pcs-Str ;
clusterP pcs-times Q ->  pcs-tol-symmetric ;
coherence
 P pcs-times Q is pcs-tol-symmetric
proof
 TolStr(# the carrier of (P pcs-times Q), the ToleranceRel of (P pcs-times Q) #) = [^P,Q^] ;
hence  P pcs-times Q is pcs-tol-symmetric by Th5; ::_thesis: verum
end;
end;

registration
let P, Q be pcs-compatible pcs-Str ;
clusterP pcs-times Q ->  pcs-compatible ;
coherence
 P pcs-times Q is pcs-compatible
proof
set R = P pcs-times Q;
set TR = the ToleranceRel of (P pcs-times Q);
set D1 = the carrier of P;
set D2 = the carrier of Q;
let p, p9, q, q9 be Element of (P pcs-times Q); :: according to PCS_0:def_22 ::_thesis: ( p (--) q & p9 <= p & q9 <= q implies p9 (--) q9 )
assume that
A1: p (--) q and
A2: p9 <= p and
A3: q9 <= q ; ::_thesis: p9 (--) q9
A4: [p,q] in the ToleranceRel of (P pcs-times Q) by A1, Def7;
then consider a, b, c, d being set  such that
A5: p = [a,b] and
A6: q = [c,d] and
A7: a in the carrier of P and
A8: b in the carrier of Q and
A9: c in the carrier of P and
A10: d in the carrier of Q and
 ( [a,c] in the ToleranceRel of P or [b,d] in the ToleranceRel of Q ) by Def2;
A11: [p9,p] in the InternalRel of (P pcs-times Q) by A2, ORDERS_2:def_5;
then  p9 in the carrier of (P pcs-times Q) by ZFMISC_1:87;
then consider e, f being set  such that
A12: e in the carrier of P and
A13: f in the carrier of Q and
A14: p9 = [e,f] by ZFMISC_1:def_2;
A15: [q9,q] in the InternalRel of (P pcs-times Q) by A3, ORDERS_2:def_5;
then  q9 in the carrier of (P pcs-times Q) by ZFMISC_1:87;
then consider g, h being set  such that
A16: g in the carrier of P and
A17: h in the carrier of Q and
A18: q9 = [g,h] by ZFMISC_1:def_2;
reconsider P = P, Q = Q as non empty pcs-compatible pcs-Str by A7, A8;
reconsider a = a, c = c, e = e, g = g as Element of P by A7, A9, A12, A16;
reconsider b = b, d = d, f = f, h = h as Element of Q by A8, A10, A13, A17;
 [^a,b^] (--) [^c,d^] by A4, A5, A6, Def7;
then A19: ( a (--) c or b (--) d ) by Th6;
A20: RelStr(# the carrier of (P pcs-times Q), the InternalRel of (P pcs-times Q) #) = [:P,Q:] by YELLOW_3:def_2;
then A21: [e,f] <= [a,b] by A5, A11, A14, ORDERS_2:def_5;
then A22: e <= a by YELLOW_3:11;
A23: f <= b by A21, YELLOW_3:11;
A24: [g,h] <= [c,d] by A6, A15, A18, A20, ORDERS_2:def_5;
then A25: g <= c by YELLOW_3:11;
 h <= d by A24, YELLOW_3:11;
then  ( e (--) g or f (--) h ) by A19, A22, A23, A25, Def22;
then A26: ( [e,g] in the ToleranceRel of P or [f,h] in the ToleranceRel of Q ) by Def7;
A27: p9 = [e,f] by A14;
 q9 = [g,h] by A18;
hence  [p9,q9] in the ToleranceRel of (P pcs-times Q) by A26, A27, Def3; :: according to PCS_0:def_7 ::_thesis: verum
end;
end;

definition
let P, Q be pcs-Str ;
funcP --> Q ->  pcs-Str  equals :: PCS_0:def 42
(pcs-reverse P) pcs-times Q;
coherence
 (pcs-reverse P) pcs-times Q is pcs-Str ;
end;

:: deftheorem  defines --> PCS_0:def_42_:_
 for P, Q being pcs-Str holds P --> Q = (pcs-reverse P) pcs-times Q;

registration
let P, Q be pcs-Str ;
clusterP --> Q ->  strict ;
coherence
 P --> Q is strict ;
end;

registration
let P, Q be non empty pcs-Str ;
clusterP --> Q ->  non empty ;
coherence
 not P --> Q is empty ;
end;

theorem :: PCS_0:39
for P, Q being pcs-Str
for p, q being Element of (P --> Q)
for p1, p2 being Element of P
for q1, q2 being Element of Q st p = [p1,q1] & q = [p2,q2] holds
( p <= q iff ( p2 <= p1 & q1 <= q2 ) )
proof
let P, Q be pcs-Str ; ::_thesis: for p, q being Element of (P --> Q)
for p1, p2 being Element of P
for q1, q2 being Element of Q st p = [p1,q1] & q = [p2,q2] holds
( p <= q iff ( p2 <= p1 & q1 <= q2 ) )
let p, q be Element of (P --> Q); ::_thesis: for p1, p2 being Element of P
for q1, q2 being Element of Q st p = [p1,q1] & q = [p2,q2] holds
( p <= q iff ( p2 <= p1 & q1 <= q2 ) )
let p1, p2 be Element of P; ::_thesis: for q1, q2 being Element of Q st p = [p1,q1] & q = [p2,q2] holds
( p <= q iff ( p2 <= p1 & q1 <= q2 ) )
let q1, q2 be Element of Q; ::_thesis: ( p = [p1,q1] & q = [p2,q2] implies ( p <= q iff ( p2 <= p1 & q1 <= q2 ) ) )
assume that
A1: p = [p1,q1] and
A2: q = [p2,q2] ; ::_thesis: ( p <= q iff ( p2 <= p1 & q1 <= q2 ) )
reconsider r1 = p1, r2 = p2 as Element of (pcs-reverse P) by Def40;
thus  ( p <= q implies ( p2 <= p1 & q1 <= q2 ) ) ::_thesis: ( p2 <= p1 & q1 <= q2 implies p <= q )
proof
assume  p <= q ; ::_thesis: ( p2 <= p1 & q1 <= q2 )
then  [p,q] in [" the InternalRel of (pcs-reverse P), the InternalRel of Q"] by ORDERS_2:def_5;
then consider a, b, s, t being set  such that
A3: p = [a,b] and
A4: q = [s,t] and
A5: [a,s] in the InternalRel of (pcs-reverse P) and
A6: [b,t] in the InternalRel of Q by YELLOW_3:def_1;
A7: a = p1 by A1, A3, XTUPLE_0:1;
A8: b = q1 by A1, A3, XTUPLE_0:1;
 s = p2 by A2, A4, XTUPLE_0:1;
then  r1 <= r2 by A5, A7, ORDERS_2:def_5;
hence  p2 <= p1 by Th33; ::_thesis: q1 <= q2
thus  [q1,q2] in the InternalRel of Q by A2, A4, A6, A8, XTUPLE_0:1; :: according to ORDERS_2:def_5 ::_thesis: verum
end;
assume that
A9: p2 <= p1 and
A10: q1 <= q2 ; ::_thesis: p <= q
 r1 <= r2 by A9, Th33;
then A11: [r1,r2] in the InternalRel of (pcs-reverse P) by ORDERS_2:def_5;
 [q1,q2] in the InternalRel of Q by A10, ORDERS_2:def_5;
hence  [p,q] in the InternalRel of (P --> Q) by A1, A2, A11, YELLOW_3:def_1; :: according to ORDERS_2:def_5 ::_thesis: verum
end;

theorem :: PCS_0:40
for P, Q being pcs-Str
for p, q being Element of (P --> Q)
for p1, p2 being Element of P
for q1, q2 being Element of Q st p = [p1,q1] & q = [p2,q2] & p (--) q & p1 (--) p2 holds
q1 (--) q2
proof
let P, Q be pcs-Str ; ::_thesis: for p, q being Element of (P --> Q)
for p1, p2 being Element of P
for q1, q2 being Element of Q st p = [p1,q1] & q = [p2,q2] & p (--) q & p1 (--) p2 holds
q1 (--) q2
let p, q be Element of (P --> Q); ::_thesis: for p1, p2 being Element of P
for q1, q2 being Element of Q st p = [p1,q1] & q = [p2,q2] & p (--) q & p1 (--) p2 holds
q1 (--) q2
let p1, p2 be Element of P; ::_thesis: for q1, q2 being Element of Q st p = [p1,q1] & q = [p2,q2] & p (--) q & p1 (--) p2 holds
q1 (--) q2
let q1, q2 be Element of Q; ::_thesis: ( p = [p1,q1] & q = [p2,q2] & p (--) q & p1 (--) p2 implies q1 (--) q2 )
assume that
A1: p = [p1,q1] and
A2: q = [p2,q2] ; ::_thesis: ( not p (--) q or not p1 (--) p2 or q1 (--) q2 )
reconsider r1 = p1, r2 = p2 as Element of (pcs-reverse P) by Def40;
assume  [p,q] in the ToleranceRel of (P --> Q) ; :: according to PCS_0:def_7 ::_thesis: ( not p1 (--) p2 or q1 (--) q2 )
then consider a, b, s, t being set  such that
A3: p = [a,b] and
A4: q = [s,t] and
 a in the carrier of (pcs-reverse P) and
 b in the carrier of Q and
 s in the carrier of (pcs-reverse P) and
 t in the carrier of Q and
A5: ( [a,s] in the ToleranceRel of (pcs-reverse P) or [b,t] in the ToleranceRel of Q ) by Def2;
A6: a = p1 by A1, A3, XTUPLE_0:1;
A7: b = q1 by A1, A3, XTUPLE_0:1;
A8: s = p2 by A2, A4, XTUPLE_0:1;
 t = q2 by A2, A4, XTUPLE_0:1;
then  ( r1 (--) r2 or q1 (--) q2 ) by A5, A6, A7, A8, Def7;
hence  ( not p1 (--) p2 or q1 (--) q2 ) by Th34; ::_thesis: verum
end;

theorem :: PCS_0:41
for P, Q being non empty pcs-Str
for p, q being Element of (P --> Q)
for p1, p2 being Element of P
for q1, q2 being Element of Q st p = [p1,q1] & q = [p2,q2] & ( not p1 (--) p2 or q1 (--) q2 ) holds
p (--) q
proof
let P, Q be non empty pcs-Str ; ::_thesis: for p, q being Element of (P --> Q)
for p1, p2 being Element of P
for q1, q2 being Element of Q st p = [p1,q1] & q = [p2,q2] & ( not p1 (--) p2 or q1 (--) q2 ) holds
p (--) q
let p, q be Element of (P --> Q); ::_thesis: for p1, p2 being Element of P
for q1, q2 being Element of Q st p = [p1,q1] & q = [p2,q2] & ( not p1 (--) p2 or q1 (--) q2 ) holds
p (--) q
let p1, p2 be Element of P; ::_thesis: for q1, q2 being Element of Q st p = [p1,q1] & q = [p2,q2] & ( not p1 (--) p2 or q1 (--) q2 ) holds
p (--) q
let q1, q2 be Element of Q; ::_thesis: ( p = [p1,q1] & q = [p2,q2] & ( not p1 (--) p2 or q1 (--) q2 ) implies p (--) q )
assume that
A1: p = [p1,q1] and
A2: q = [p2,q2] ; ::_thesis: ( ( p1 (--) p2 & not q1 (--) q2 ) or p (--) q )
reconsider r1 = p1, r2 = p2 as Element of (pcs-reverse P) by Def40;
reconsider w1 = [r1,q1], w2 = [r2,q2] as Element of ((pcs-reverse P) pcs-times Q) by A1, A2;
assume  ( not p1 (--) p2 or q1 (--) q2 ) ; ::_thesis: p (--) q
then  ( r1 (--) r2 or q1 (--) q2 ) by Th35;
then  w1 (--) w2 by Th38;
hence  p (--) q by A1, A2; ::_thesis: verum
end;

registration
let P, Q be reflexive pcs-Str ;
clusterP --> Q ->  reflexive ;
coherence
 P --> Q is reflexive ;
end;

registration
let P, Q be transitive pcs-Str ;
clusterP --> Q ->  transitive ;
coherence
 P --> Q is transitive ;
end;

registration
let P be pcs-Str ;
let Q be pcs-tol-reflexive pcs-Str ;
clusterP --> Q ->  pcs-tol-reflexive ;
coherence
 P --> Q is pcs-tol-reflexive ;
end;

registration
let P be pcs-tol-irreflexive pcs-Str ;
let Q be pcs-Str ;
clusterP --> Q ->  pcs-tol-reflexive ;
coherence
 P --> Q is pcs-tol-reflexive ;
end;

registration
let P, Q be pcs-tol-symmetric pcs-Str ;
clusterP --> Q ->  pcs-tol-symmetric ;
coherence
 P --> Q is pcs-tol-symmetric ;
end;

registration
let P, Q be pcs-compatible pcs-Str ;
clusterP --> Q ->  pcs-compatible ;
coherence
 P --> Q is pcs-compatible ;
end;

registration
let P, Q be pcs;
clusterP --> Q ->  pcs-like ;
coherence
 P --> Q is pcs-like ;
end;

definition
let P be TolStr ;
let S be Subset of P;
attrS is pcs-self-coherent  means :Def43: :: PCS_0:def 43
 for x, y being Element of P st x in S & y in S holds
x (--) y;
end;

:: deftheorem Def43 defines pcs-self-coherent PCS_0:def_43_:_
 for P being TolStr
for S being Subset of P holds
( S is pcs-self-coherent iff for x, y being Element of P st x in S & y in S holds
x (--) y );

registration
let P be TolStr ;
cluster empty  ->  pcs-self-coherent for Element of bool the carrier of P;
coherence
 for b1 being Subset of P st b1 is empty holds
b1 is pcs-self-coherent
proof
let S be Subset of P; ::_thesis: ( S is empty implies S is pcs-self-coherent )
assume A1: S is empty ; ::_thesis: S is pcs-self-coherent
let x be Element of P; :: according to PCS_0:def_43 ::_thesis: for y being Element of P st x in S & y in S holds
x (--) y
thus  for y being Element of P st x in S & y in S holds
x (--) y by A1; ::_thesis: verum
end;
end;

definition
let P be TolStr ;
let F be Subset-Family of P;
attrF is pcs-self-coherent-membered  means :Def44: :: PCS_0:def 44
 for S being Subset of P st S in F holds
S is pcs-self-coherent ;
end;

:: deftheorem Def44 defines pcs-self-coherent-membered PCS_0:def_44_:_
 for P being TolStr
for F being Subset-Family of P holds
( F is pcs-self-coherent-membered iff for S being Subset of P st S in F holds
S is pcs-self-coherent );

registration
let P be TolStr ;
cluster non empty pcs-self-coherent-membered for Element of bool (bool the carrier of P);
existence
 ex b1 being Subset-Family of P st
( not b1 is empty & b1 is pcs-self-coherent-membered )
proof
reconsider F = {{}} as Subset-Family of P by SETFAM_1:46;
take  F ; ::_thesis: ( not F is empty & F is pcs-self-coherent-membered )
thus  not F is empty ; ::_thesis: F is pcs-self-coherent-membered
let S be Subset of P; :: according to PCS_0:def_44 ::_thesis: ( S in F implies S is pcs-self-coherent )
assume  S in F ; ::_thesis: S is pcs-self-coherent
then  S = {} P by TARSKI:def_1;
hence  S is pcs-self-coherent ; ::_thesis: verum
end;
end;

definition
let P be RelStr ;
let D be set ;
defpred S1[ set , set ] means ( $1 in D & $2 in D & ( for a being set st a in $1 holds
ex b being set st
( b in $2 & [a,b] in the InternalRel of P ) ) );
func pcs-general-power-IR (P,D) ->  Relation of D means :Def45: :: PCS_0:def 45
 for A, B being set holds
( [A,B] in it iff ( A in D & B in D & ( for a being set st a in A holds
ex b being set st
( b in B & [a,b] in the InternalRel of P ) ) ) );
existence
 ex b1 being Relation of D st
for A, B being set holds
( [A,B] in b1 iff ( A in D & B in D & ( for a being set st a in A holds
ex b being set st
( b in B & [a,b] in the InternalRel of P ) ) ) )
proof
consider R being Relation of D such that
A1: for x, y being set holds
( [x,y] in R iff ( x in D & y in D & S1[x,y] ) ) from RELSET_1:sch_1();
take  R ; ::_thesis: for A, B being set holds
( [A,B] in R iff ( A in D & B in D & ( for a being set st a in A holds
ex b being set st
( b in B & [a,b] in the InternalRel of P ) ) ) )
thus  for A, B being set holds
( [A,B] in R iff ( A in D & B in D & ( for a being set st a in A holds
ex b being set st
( b in B & [a,b] in the InternalRel of P ) ) ) ) by A1; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Relation of D st ( for A, B being set holds
( [A,B] in b1 iff ( A in D & B in D & ( for a being set st a in A holds
ex b being set st
( b in B & [a,b] in the InternalRel of P ) ) ) ) ) & ( for A, B being set holds
( [A,B] in b2 iff ( A in D & B in D & ( for a being set st a in A holds
ex b being set st
( b in B & [a,b] in the InternalRel of P ) ) ) ) ) holds
b1 = b2
proof
let R1, R2 be Relation of D; ::_thesis: ( ( for A, B being set holds
( [A,B] in R1 iff ( A in D & B in D & ( for a being set st a in A holds
ex b being set st
( b in B & [a,b] in the InternalRel of P ) ) ) ) ) & ( for A, B being set holds
( [A,B] in R2 iff ( A in D & B in D & ( for a being set st a in A holds
ex b being set st
( b in B & [a,b] in the InternalRel of P ) ) ) ) ) implies R1 = R2 )
assume that
A2: for A, B being set holds
( [A,B] in R1 iff ( A in D & B in D & ( for a being set st a in A holds
ex b being set st
( b in B & [a,b] in the InternalRel of P ) ) ) ) and
A3: for A, B being set holds
( [A,B] in R2 iff ( A in D & B in D & ( for a being set st a in A holds
ex b being set st
( b in B & [a,b] in the InternalRel of P ) ) ) ) ; ::_thesis: R1 = R2
A4: for a, b being set holds
( [a,b] in R1 iff S1[a,b] ) by A2;
A5: for a, b being set holds
( [a,b] in R2 iff S1[a,b] ) by A3;
thus  R1 = R2 from RELAT_1:sch_2(A4, A5); ::_thesis: verum
end;
end;

:: deftheorem Def45 defines pcs-general-power-IR PCS_0:def_45_:_
 for P being RelStr
for D being set
for b3 being Relation of D holds
( b3 = pcs-general-power-IR (P,D) iff for A, B being set holds
( [A,B] in b3 iff ( A in D & B in D & ( for a being set st a in A holds
ex b being set st
( b in B & [a,b] in the InternalRel of P ) ) ) ) );

definition
let P be TolStr ;
let D be set ;
defpred S1[ set , set ] means ( $1 in D & $2 in D & ( for a, b being set st a in $1 & b in $2 holds
[a,b] in the ToleranceRel of P ) );
func pcs-general-power-TR (P,D) ->  Relation of D means :Def46: :: PCS_0:def 46
 for A, B being set holds
( [A,B] in it iff ( A in D & B in D & ( for a, b being set st a in A & b in B holds
[a,b] in the ToleranceRel of P ) ) );
existence
 ex b1 being Relation of D st
for A, B being set holds
( [A,B] in b1 iff ( A in D & B in D & ( for a, b being set st a in A & b in B holds
[a,b] in the ToleranceRel of P ) ) )
proof
consider R being Relation of D such that
A1: for x, y being set holds
( [x,y] in R iff ( x in D & y in D & S1[x,y] ) ) from RELSET_1:sch_1();
take  R ; ::_thesis: for A, B being set holds
( [A,B] in R iff ( A in D & B in D & ( for a, b being set st a in A & b in B holds
[a,b] in the ToleranceRel of P ) ) )
thus  for A, B being set holds
( [A,B] in R iff ( A in D & B in D & ( for a, b being set st a in A & b in B holds
[a,b] in the ToleranceRel of P ) ) ) by A1; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Relation of D st ( for A, B being set holds
( [A,B] in b1 iff ( A in D & B in D & ( for a, b being set st a in A & b in B holds
[a,b] in the ToleranceRel of P ) ) ) ) & ( for A, B being set holds
( [A,B] in b2 iff ( A in D & B in D & ( for a, b being set st a in A & b in B holds
[a,b] in the ToleranceRel of P ) ) ) ) holds
b1 = b2
proof
let R1, R2 be Relation of D; ::_thesis: ( ( for A, B being set holds
( [A,B] in R1 iff ( A in D & B in D & ( for a, b being set st a in A & b in B holds
[a,b] in the ToleranceRel of P ) ) ) ) & ( for A, B being set holds
( [A,B] in R2 iff ( A in D & B in D & ( for a, b being set st a in A & b in B holds
[a,b] in the ToleranceRel of P ) ) ) ) implies R1 = R2 )
assume that
A2: for A, B being set holds
( [A,B] in R1 iff ( A in D & B in D & ( for a, b being set st a in A & b in B holds
[a,b] in the ToleranceRel of P ) ) ) and
A3: for A, B being set holds
( [A,B] in R2 iff ( A in D & B in D & ( for a, b being set st a in A & b in B holds
[a,b] in the ToleranceRel of P ) ) ) ; ::_thesis: R1 = R2
A4: for a, b being set holds
( [a,b] in R1 iff S1[a,b] ) by A2;
A5: for a, b being set holds
( [a,b] in R2 iff S1[a,b] ) by A3;
thus  R1 = R2 from RELAT_1:sch_2(A4, A5); ::_thesis: verum
end;
end;

:: deftheorem Def46 defines pcs-general-power-TR PCS_0:def_46_:_
 for P being TolStr
for D being set
for b3 being Relation of D holds
( b3 = pcs-general-power-TR (P,D) iff for A, B being set holds
( [A,B] in b3 iff ( A in D & B in D & ( for a, b being set st a in A & b in B holds
[a,b] in the ToleranceRel of P ) ) ) );

theorem :: PCS_0:42
for P being RelStr
for D being Subset-Family of P
for A, B being set holds
( [A,B] in pcs-general-power-IR (P,D) iff ( A in D & B in D & ( for a being Element of P st a in A holds
ex b being Element of P st
( b in B & a <= b ) ) ) )
proof
let P be RelStr ; ::_thesis: for D being Subset-Family of P
for A, B being set holds
( [A,B] in pcs-general-power-IR (P,D) iff ( A in D & B in D & ( for a being Element of P st a in A holds
ex b being Element of P st
( b in B & a <= b ) ) ) )
let D be Subset-Family of P; ::_thesis: for A, B being set holds
( [A,B] in pcs-general-power-IR (P,D) iff ( A in D & B in D & ( for a being Element of P st a in A holds
ex b being Element of P st
( b in B & a <= b ) ) ) )
let A, B be set ; ::_thesis: ( [A,B] in pcs-general-power-IR (P,D) iff ( A in D & B in D & ( for a being Element of P st a in A holds
ex b being Element of P st
( b in B & a <= b ) ) ) )
thus  ( [A,B] in pcs-general-power-IR (P,D) implies ( A in D & B in D & ( for a being Element of P st a in A holds
ex b being Element of P st
( b in B & a <= b ) ) ) ) ::_thesis: ( A in D & B in D & ( for a being Element of P st a in A holds
ex b being Element of P st
( b in B & a <= b ) ) implies [A,B] in pcs-general-power-IR (P,D) )
proof
assume A1: [A,B] in pcs-general-power-IR (P,D) ; ::_thesis: ( A in D & B in D & ( for a being Element of P st a in A holds
ex b being Element of P st
( b in B & a <= b ) ) )
hence A2: ( A in D & B in D ) by Def45; ::_thesis: for a being Element of P st a in A holds
ex b being Element of P st
( b in B & a <= b )
let a be Element of P; ::_thesis: ( a in A implies ex b being Element of P st
( b in B & a <= b ) )
assume  a in A ; ::_thesis: ex b being Element of P st
( b in B & a <= b )
then consider b being set  such that
A3: b in B and
A4: [a,b] in the InternalRel of P by A1, Def45;
reconsider b = b as Element of P by A2, A3;
take  b ; ::_thesis: ( b in B & a <= b )
thus  ( b in B & a <= b ) by A3, A4, ORDERS_2:def_5; ::_thesis: verum
end;
assume that
A5: A in D and
A6: B in D and
A7: for a being Element of P st a in A holds
ex b being Element of P st
( b in B & a <= b ) ; ::_thesis: [A,B] in pcs-general-power-IR (P,D)
 for a being set st a in A holds
ex b being set st
( b in B & [a,b] in the InternalRel of P )
proof
let a be set ; ::_thesis: ( a in A implies ex b being set st
( b in B & [a,b] in the InternalRel of P ) )
assume A8: a in A ; ::_thesis: ex b being set st
( b in B & [a,b] in the InternalRel of P )
then reconsider a = a as Element of P by A5;
consider b being Element of P such that
A9: b in B and
A10: a <= b by A7, A8;
take  b ; ::_thesis: ( b in B & [a,b] in the InternalRel of P )
thus  ( b in B & [a,b] in the InternalRel of P ) by A9, A10, ORDERS_2:def_5; ::_thesis: verum
end;
hence  [A,B] in pcs-general-power-IR (P,D) by A5, A6, Def45; ::_thesis: verum
end;

theorem :: PCS_0:43
for P being TolStr
for D being Subset-Family of P
for A, B being set holds
( [A,B] in pcs-general-power-TR (P,D) iff ( A in D & B in D & ( for a, b being Element of P st a in A & b in B holds
a (--) b ) ) )
proof
let P be TolStr ; ::_thesis: for D being Subset-Family of P
for A, B being set holds
( [A,B] in pcs-general-power-TR (P,D) iff ( A in D & B in D & ( for a, b being Element of P st a in A & b in B holds
a (--) b ) ) )
let D be Subset-Family of P; ::_thesis: for A, B being set holds
( [A,B] in pcs-general-power-TR (P,D) iff ( A in D & B in D & ( for a, b being Element of P st a in A & b in B holds
a (--) b ) ) )
let A, B be set ; ::_thesis: ( [A,B] in pcs-general-power-TR (P,D) iff ( A in D & B in D & ( for a, b being Element of P st a in A & b in B holds
a (--) b ) ) )
thus  ( [A,B] in pcs-general-power-TR (P,D) implies ( A in D & B in D & ( for a, b being Element of P st a in A & b in B holds
a (--) b ) ) ) ::_thesis: ( A in D & B in D & ( for a, b being Element of P st a in A & b in B holds
a (--) b ) implies [A,B] in pcs-general-power-TR (P,D) )
proof
assume A1: [A,B] in pcs-general-power-TR (P,D) ; ::_thesis: ( A in D & B in D & ( for a, b being Element of P st a in A & b in B holds
a (--) b ) )
hence  ( A in D & B in D ) by Def46; ::_thesis: for a, b being Element of P st a in A & b in B holds
a (--) b
let a, b be Element of P; ::_thesis: ( a in A & b in B implies a (--) b )
assume that
A2: a in A and
A3: b in B ; ::_thesis: a (--) b
 [a,b] in the ToleranceRel of P by A1, A2, A3, Def46;
hence  a (--) b by Def7; ::_thesis: verum
end;
assume that
A4: A in D and
A5: B in D and
A6: for a, b being Element of P st a in A & b in B holds
a (--) b ; ::_thesis: [A,B] in pcs-general-power-TR (P,D)
 for a, b being set st a in A & b in B holds
[a,b] in the ToleranceRel of P
proof
let a, b be set ; ::_thesis: ( a in A & b in B implies [a,b] in the ToleranceRel of P )
assume that
A7: a in A and
A8: b in B ; ::_thesis: [a,b] in the ToleranceRel of P
reconsider a = a, b = b as Element of P by A4, A5, A7, A8;
 a (--) b by A6, A7, A8;
hence  [a,b] in the ToleranceRel of P by Def7; ::_thesis: verum
end;
hence  [A,B] in pcs-general-power-TR (P,D) by A4, A5, Def46; ::_thesis: verum
end;

definition
let P be pcs-Str ;
let D be set ;
func pcs-general-power (P,D) ->  pcs-Str  equals :: PCS_0:def 47
 pcs-Str(# D,(pcs-general-power-IR (P,D)),(pcs-general-power-TR (P,D)) #);
coherence
 pcs-Str(# D,(pcs-general-power-IR (P,D)),(pcs-general-power-TR (P,D)) #) is pcs-Str ;
end;

:: deftheorem  defines pcs-general-power PCS_0:def_47_:_
 for P being pcs-Str
for D being set holds pcs-general-power (P,D) = pcs-Str(# D,(pcs-general-power-IR (P,D)),(pcs-general-power-TR (P,D)) #);

notation
let P be pcs-Str ;
let D be Subset-Family of P;
synonym  pcs-general-power D for  pcs-general-power (P,D);
end;

registration
let P be pcs-Str ;
let D be non empty set ;
cluster pcs-general-power (P,D) ->  non empty ;
coherence
 not pcs-general-power (P,D) is empty ;
end;

theorem Th44: :: PCS_0:44
for P being pcs-Str
for D being set
for p, q being Element of (pcs-general-power (P,D)) st p <= q holds
for p9 being Element of P st p9 in p holds
ex q9 being Element of P st
( q9 in q & p9 <= q9 )
proof
let P be pcs-Str ; ::_thesis: for D being set
for p, q being Element of (pcs-general-power (P,D)) st p <= q holds
for p9 being Element of P st p9 in p holds
ex q9 being Element of P st
( q9 in q & p9 <= q9 )
let D be set ; ::_thesis: for p, q being Element of (pcs-general-power (P,D)) st p <= q holds
for p9 being Element of P st p9 in p holds
ex q9 being Element of P st
( q9 in q & p9 <= q9 )
set R = pcs-general-power (P,D);
let p, q be Element of (pcs-general-power (P,D)); ::_thesis: ( p <= q implies for p9 being Element of P st p9 in p holds
ex q9 being Element of P st
( q9 in q & p9 <= q9 ) )
assume A1: [p,q] in the InternalRel of (pcs-general-power (P,D)) ; :: according to ORDERS_2:def_5 ::_thesis: for p9 being Element of P st p9 in p holds
ex q9 being Element of P st
( q9 in q & p9 <= q9 )
let p9 be Element of P; ::_thesis: ( p9 in p implies ex q9 being Element of P st
( q9 in q & p9 <= q9 ) )
assume  p9 in p ; ::_thesis: ex q9 being Element of P st
( q9 in q & p9 <= q9 )
then consider b being set  such that
A2: b in q and
A3: [p9,b] in the InternalRel of P by A1, Def45;
reconsider b = b as Element of P by A3, ZFMISC_1:87;
take  b ; ::_thesis: ( b in q & p9 <= b )
thus  ( b in q & [p9,b] in the InternalRel of P ) by A2, A3; :: according to ORDERS_2:def_5 ::_thesis: verum
end;

theorem :: PCS_0:45
for P being pcs-Str
for D being non empty Subset-Family of P
for p, q being Element of (pcs-general-power D) st ( for p9 being Element of P st p9 in p holds
ex q9 being Element of P st
( q9 in q & p9 <= q9 ) ) holds
p <= q
proof
let P be pcs-Str ; ::_thesis: for D being non empty Subset-Family of P
for p, q being Element of (pcs-general-power D) st ( for p9 being Element of P st p9 in p holds
ex q9 being Element of P st
( q9 in q & p9 <= q9 ) ) holds
p <= q
let D be non empty Subset-Family of P; ::_thesis: for p, q being Element of (pcs-general-power D) st ( for p9 being Element of P st p9 in p holds
ex q9 being Element of P st
( q9 in q & p9 <= q9 ) ) holds
p <= q
set R = pcs-general-power D;
let p, q be Element of (pcs-general-power D); ::_thesis: ( ( for p9 being Element of P st p9 in p holds
ex q9 being Element of P st
( q9 in q & p9 <= q9 ) ) implies p <= q )
assume A1: for p9 being Element of P st p9 in p holds
ex q9 being Element of P st
( q9 in q & p9 <= q9 ) ; ::_thesis: p <= q
A2: p in D ;
 for a being set st a in p holds
ex b being set st
( b in q & [a,b] in the InternalRel of P )
proof
let a be set ; ::_thesis: ( a in p implies ex b being set st
( b in q & [a,b] in the InternalRel of P ) )
assume A3: a in p ; ::_thesis: ex b being set st
( b in q & [a,b] in the InternalRel of P )
then reconsider a = a as Element of P by A2;
consider q9 being Element of P such that
A4: q9 in q and
A5: a <= q9 by A1, A3;
take  q9 ; ::_thesis: ( q9 in q & [a,q9] in the InternalRel of P )
thus  ( q9 in q & [a,q9] in the InternalRel of P ) by A4, A5, ORDERS_2:def_5; ::_thesis: verum
end;
hence  [p,q] in the InternalRel of (pcs-general-power D) by Def45; :: according to ORDERS_2:def_5 ::_thesis: verum
end;

theorem Th46: :: PCS_0:46
for P being pcs-Str
for D being set
for p, q being Element of (pcs-general-power (P,D)) st p (--) q holds
for p9, q9 being Element of P st p9 in p & q9 in q holds
p9 (--) q9
proof
let P be pcs-Str ; ::_thesis: for D being set
for p, q being Element of (pcs-general-power (P,D)) st p (--) q holds
for p9, q9 being Element of P st p9 in p & q9 in q holds
p9 (--) q9
let D be set ; ::_thesis: for p, q being Element of (pcs-general-power (P,D)) st p (--) q holds
for p9, q9 being Element of P st p9 in p & q9 in q holds
p9 (--) q9
set R = pcs-general-power (P,D);
let p, q be Element of (pcs-general-power (P,D)); ::_thesis: ( p (--) q implies for p9, q9 being Element of P st p9 in p & q9 in q holds
p9 (--) q9 )
assume A1: [p,q] in the ToleranceRel of (pcs-general-power (P,D)) ; :: according to PCS_0:def_7 ::_thesis: for p9, q9 being Element of P st p9 in p & q9 in q holds
p9 (--) q9
let p9, q9 be Element of P; ::_thesis: ( p9 in p & q9 in q implies p9 (--) q9 )
assume that
A2: p9 in p and
A3: q9 in q ; ::_thesis: p9 (--) q9
thus  [p9,q9] in the ToleranceRel of P by A1, A2, A3, Def46; :: according to PCS_0:def_7 ::_thesis: verum
end;

theorem :: PCS_0:47
for P being pcs-Str
for D being non empty Subset-Family of P
for p, q being Element of (pcs-general-power D) st ( for p9, q9 being Element of P st p9 in p & q9 in q holds
p9 (--) q9 ) holds
p (--) q
proof
let P be pcs-Str ; ::_thesis: for D being non empty Subset-Family of P
for p, q being Element of (pcs-general-power D) st ( for p9, q9 being Element of P st p9 in p & q9 in q holds
p9 (--) q9 ) holds
p (--) q
let D be non empty Subset-Family of P; ::_thesis: for p, q being Element of (pcs-general-power D) st ( for p9, q9 being Element of P st p9 in p & q9 in q holds
p9 (--) q9 ) holds
p (--) q
set R = pcs-general-power D;
let p, q be Element of (pcs-general-power D); ::_thesis: ( ( for p9, q9 being Element of P st p9 in p & q9 in q holds
p9 (--) q9 ) implies p (--) q )
assume A1: for p9, q9 being Element of P st p9 in p & q9 in q holds
p9 (--) q9 ; ::_thesis: p (--) q
A2: p in D ;
A3: q in D ;
now__::_thesis:_for_a,_b_being_set_st_a_in_p_&_b_in_q_holds_
[a,b]_in_the_ToleranceRel_of_P
let a, b be set ; ::_thesis: ( a in p & b in q implies [a,b] in the ToleranceRel of P )
assume that
A4: a in p and
A5: b in q ; ::_thesis: [a,b] in the ToleranceRel of P
reconsider a1 = a, b1 = b as Element of P by A2, A3, A4, A5;
 a1 (--) b1 by A1, A4, A5;
hence  [a,b] in the ToleranceRel of P by Def7; ::_thesis: verum
end;
hence  [p,q] in the ToleranceRel of (pcs-general-power D) by Def46; :: according to PCS_0:def_7 ::_thesis: verum
end;

registration
let P be pcs-Str ;
let D be set ;
cluster pcs-general-power (P,D) ->  strict ;
coherence
 pcs-general-power (P,D) is strict ;
end;

registration
let P be reflexive pcs-Str ;
let D be Subset-Family of P;
cluster pcs-general-power (P,D) ->  reflexive ;
coherence
 pcs-general-power D is reflexive
proof
set R = pcs-general-power D;
let x be set ; :: according to RELAT_2:def_1,ORDERS_2:def_2 ::_thesis: ( not x in the carrier of (pcs-general-power D) or [^,^] in the InternalRel of (pcs-general-power D) )
assume A1: x in the carrier of (pcs-general-power D) ; ::_thesis: [^,^] in the InternalRel of (pcs-general-power D)
 for a being set st a in x holds
ex b being set st
( b in x & [a,b] in the InternalRel of P )
proof
let a be set ; ::_thesis: ( a in x implies ex b being set st
( b in x & [a,b] in the InternalRel of P ) )
assume A2: a in x ; ::_thesis: ex b being set st
( b in x & [a,b] in the InternalRel of P )
take  a ; ::_thesis: ( a in x & [a,a] in the InternalRel of P )
thus  a in x by A2; ::_thesis: [a,a] in the InternalRel of P
 field the InternalRel of P = the carrier of P by ORDERS_1:12;
then  the InternalRel of P is_reflexive_in the carrier of P by RELAT_2:def_9;
hence  [a,a] in the InternalRel of P by A1, A2, RELAT_2:def_1; ::_thesis: verum
end;
hence  [^,^] in the InternalRel of (pcs-general-power D) by A1, Def45; ::_thesis: verum
end;
end;

registration
let P be transitive pcs-Str ;
let D be set ;
cluster pcs-general-power (P,D) ->  transitive ;
coherence
 pcs-general-power (P,D) is transitive
proof
set R = pcs-general-power (P,D);
set IR = the InternalRel of (pcs-general-power (P,D));
let x, y, z be set ; :: according to RELAT_2:def_8,ORDERS_2:def_3 ::_thesis: ( not x in the carrier of (pcs-general-power (P,D)) or not y in the carrier of (pcs-general-power (P,D)) or not z in the carrier of (pcs-general-power (P,D)) or not [^,^] in the InternalRel of (pcs-general-power (P,D)) or not [^,^] in the InternalRel of (pcs-general-power (P,D)) or [^,^] in the InternalRel of (pcs-general-power (P,D)) )
assume that
A1: x in the carrier of (pcs-general-power (P,D)) and
 y in the carrier of (pcs-general-power (P,D)) and
A2: z in the carrier of (pcs-general-power (P,D)) and
A3: [x,y] in the InternalRel of (pcs-general-power (P,D)) and
A4: [y,z] in the InternalRel of (pcs-general-power (P,D)) ; ::_thesis: [^,^] in the InternalRel of (pcs-general-power (P,D))
 for a being set st a in x holds
ex b being set st
( b in z & [a,b] in the InternalRel of P )
proof
let a be set ; ::_thesis: ( a in x implies ex b being set st
( b in z & [a,b] in the InternalRel of P ) )
assume  a in x ; ::_thesis: ex b being set st
( b in z & [a,b] in the InternalRel of P )
then consider b being set  such that
A5: b in y and
A6: [a,b] in the InternalRel of P by A3, Def45;
consider c being set  such that
A7: c in z and
A8: [b,c] in the InternalRel of P by A4, A5, Def45;
take  c ; ::_thesis: ( c in z & [a,c] in the InternalRel of P )
thus  c in z by A7; ::_thesis: [a,c] in the InternalRel of P
A9: the InternalRel of P is_transitive_in the carrier of P by ORDERS_2:def_3;
A10: a in the carrier of P by A6, ZFMISC_1:87;
A11: b in the carrier of P by A6, ZFMISC_1:87;
 c in the carrier of P by A8, ZFMISC_1:87;
hence  [a,c] in the InternalRel of P by A6, A8, A9, A10, A11, RELAT_2:def_8; ::_thesis: verum
end;
hence  [^,^] in the InternalRel of (pcs-general-power (P,D)) by A1, A2, Def45; ::_thesis: verum
end;
end;

registration
let P be pcs-tol-reflexive pcs-Str ;
let D be pcs-self-coherent-membered Subset-Family of P;
cluster pcs-general-power (P,D) ->  pcs-tol-reflexive ;
coherence
 pcs-general-power D is pcs-tol-reflexive
proof
let x be set ; :: according to RELAT_2:def_1,PCS_0:def_9 ::_thesis: ( not x in the carrier of (pcs-general-power D) or [^,^] in the ToleranceRel of (pcs-general-power D) )
assume A1: x in the carrier of (pcs-general-power D) ; ::_thesis: [^,^] in the ToleranceRel of (pcs-general-power D)
then reconsider x = x as Subset of P ;
A2: x is pcs-self-coherent by A1, Def44;
now__::_thesis:_for_a,_b_being_set_st_a_in_x_&_b_in_x_holds_
[a,b]_in_the_ToleranceRel_of_P
let a, b be set ; ::_thesis: ( a in x & b in x implies [a,b] in the ToleranceRel of P )
assume that
A3: a in x and
A4: b in x ; ::_thesis: [a,b] in the ToleranceRel of P
reconsider a1 = a, b1 = b as Element of P by A3, A4;
 a1 (--) b1 by A2, A3, A4, Def43;
hence  [a,b] in the ToleranceRel of P by Def7; ::_thesis: verum
end;
hence  [^,^] in the ToleranceRel of (pcs-general-power D) by A1, Def46; ::_thesis: verum
end;
end;

registration
let P be pcs-tol-symmetric pcs-Str ;
let D be Subset-Family of P;
cluster pcs-general-power (P,D) ->  pcs-tol-symmetric ;
coherence
 pcs-general-power D is pcs-tol-symmetric
proof
set R = pcs-general-power D;
let x, y be set ; :: according to RELAT_2:def_3,PCS_0:def_11 ::_thesis: ( not x in the carrier of (pcs-general-power D) or not y in the carrier of (pcs-general-power D) or not [^,^] in the ToleranceRel of (pcs-general-power D) or [^,^] in the ToleranceRel of (pcs-general-power D) )
assume A1: x in the carrier of (pcs-general-power D) ; ::_thesis: ( not y in the carrier of (pcs-general-power D) or not [^,^] in the ToleranceRel of (pcs-general-power D) or [^,^] in the ToleranceRel of (pcs-general-power D) )
assume A2: y in the carrier of (pcs-general-power D) ; ::_thesis: ( not [^,^] in the ToleranceRel of (pcs-general-power D) or [^,^] in the ToleranceRel of (pcs-general-power D) )
assume A3: [x,y] in the ToleranceRel of (pcs-general-power D) ; ::_thesis: [^,^] in the ToleranceRel of (pcs-general-power D)
now__::_thesis:_for_a,_b_being_set_st_a_in_y_&_b_in_x_holds_
[a,b]_in_the_ToleranceRel_of_P
let a, b be set ; ::_thesis: ( a in y & b in x implies [a,b] in the ToleranceRel of P )
assume that
A4: a in y and
A5: b in x ; ::_thesis: [a,b] in the ToleranceRel of P
reconsider a1 = a, b1 = b as Element of P by A1, A2, A4, A5;
 [b,a] in the ToleranceRel of P by A3, A4, A5, Def46;
then  b1 (--) a1 by Def7;
hence  [a,b] in the ToleranceRel of P by Def7; ::_thesis: verum
end;
hence  [^,^] in the ToleranceRel of (pcs-general-power D) by A1, A2, Def46; ::_thesis: verum
end;
end;

registration
let P be pcs-compatible pcs-Str ;
let D be Subset-Family of P;
cluster pcs-general-power (P,D) ->  pcs-compatible ;
coherence
 pcs-general-power D is pcs-compatible
proof
set R = pcs-general-power D;
let p, p9, q, q9 be Element of (pcs-general-power D); :: according to PCS_0:def_22 ::_thesis: ( p (--) q & p9 <= p & q9 <= q implies p9 (--) q9 )
assume that
A1: p (--) q and
A2: p9 <= p and
A3: q9 <= q ; ::_thesis: p9 (--) q9
A4: [p9,p] in the InternalRel of (pcs-general-power D) by A2, ORDERS_2:def_5;
A5: [q9,q] in the InternalRel of (pcs-general-power D) by A3, ORDERS_2:def_5;
A6: p9 in the carrier of (pcs-general-power D) by A4, ZFMISC_1:87;
A7: q9 in the carrier of (pcs-general-power D) by A5, ZFMISC_1:87;
now__::_thesis:_for_a,_b_being_set_st_a_in_p9_&_b_in_q9_holds_
[a,b]_in_the_ToleranceRel_of_P
let a, b be set ; ::_thesis: ( a in p9 & b in q9 implies [a,b] in the ToleranceRel of P )
assume that
A8: a in p9 and
A9: b in q9 ; ::_thesis: [a,b] in the ToleranceRel of P
reconsider a1 = a, b1 = b as Element of P by A6, A7, A8, A9;
consider p1 being Element of P such that
A10: p1 in p and
A11: a1 <= p1 by A2, A8, Th44;
consider q1 being Element of P such that
A12: q1 in q and
A13: b1 <= q1 by A3, A9, Th44;
 p1 (--) q1 by A1, A10, A12, Th46;
then  a1 (--) b1 by A11, A13, Def22;
hence  [a,b] in the ToleranceRel of P by Def7; ::_thesis: verum
end;
hence  [p9,q9] in the ToleranceRel of (pcs-general-power D) by A6, Def46; :: according to PCS_0:def_7 ::_thesis: verum
end;
end;

definition
let P be pcs-Str ;
func pcs-coherent-power P ->  set  equals :: PCS_0:def 48
 {  X where X is Subset of P : X is pcs-self-coherent  }  ;
coherence
  {  X where X is Subset of P : X is pcs-self-coherent  }  is set ;
end;

:: deftheorem  defines pcs-coherent-power PCS_0:def_48_:_
 for P being pcs-Str holds pcs-coherent-power P =  {  X where X is Subset of P : X is pcs-self-coherent  }  ;

registration
let P be pcs-Str ;
cluster pcs-self-coherent for Element of bool the carrier of P;
existence
 ex b1 being Subset of P st b1 is pcs-self-coherent
proof
take  {} P ; ::_thesis: {} P is pcs-self-coherent
thus  {} P is pcs-self-coherent ; ::_thesis: verum
end;
end;

theorem Th48: :: PCS_0:48
for P being pcs-Str
for X being set st X in pcs-coherent-power P holds
X is pcs-self-coherent Subset of P
proof
let P be pcs-Str ; ::_thesis: for X being set st X in pcs-coherent-power P holds
X is pcs-self-coherent Subset of P
let X be set ; ::_thesis: ( X in pcs-coherent-power P implies X is pcs-self-coherent Subset of P )
assume  X in pcs-coherent-power P ; ::_thesis: X is pcs-self-coherent Subset of P
then  ex Y being Subset of P st
( X = Y & Y is pcs-self-coherent ) ;
hence  X is pcs-self-coherent Subset of P ; ::_thesis: verum
end;

registration
let P be pcs-Str ;
cluster pcs-coherent-power P ->  non empty ;
coherence
 not pcs-coherent-power P is empty
proof
 {} P in pcs-coherent-power P ;
hence  not pcs-coherent-power P is empty ; ::_thesis: verum
end;
end;

definition
let P be pcs-Str ;
:: original: pcs-coherent-power
redefine func pcs-coherent-power P ->  Subset-Family of P;
coherence
 pcs-coherent-power P is Subset-Family of P
proof
 pcs-coherent-power P c= bool the carrier of P
proof
let X be set ; :: according to TARSKI:def_3 ::_thesis: ( not X in pcs-coherent-power P or X in bool the carrier of P )
assume  X in pcs-coherent-power P ; ::_thesis: X in bool the carrier of P
then  X is Subset of P by Th48;
hence  X in bool the carrier of P ; ::_thesis: verum
end;
hence  pcs-coherent-power P is Subset-Family of P ; ::_thesis: verum
end;
end;

registration
let P be pcs-Str ;
cluster pcs-coherent-power P ->  pcs-self-coherent-membered for Subset-Family of P;
coherence
 for b1 being Subset-Family of P st b1 = pcs-coherent-power P holds
b1 is pcs-self-coherent-membered
proof
 pcs-coherent-power P is pcs-self-coherent-membered
proof
let S be Subset of P; :: according to PCS_0:def_44 ::_thesis: ( S in pcs-coherent-power P implies S is pcs-self-coherent )
thus  ( S in pcs-coherent-power P implies S is pcs-self-coherent ) by Th48; ::_thesis: verum
end;
hence  for b1 being Subset-Family of P st b1 = pcs-coherent-power P holds
b1 is pcs-self-coherent-membered ; ::_thesis: verum
end;
end;

definition
let P be pcs-Str ;
func pcs-power P ->  pcs-Str  equals :: PCS_0:def 49
 pcs-general-power (pcs-coherent-power P);
coherence
 pcs-general-power (pcs-coherent-power P) is pcs-Str ;
end;

:: deftheorem  defines pcs-power PCS_0:def_49_:_
 for P being pcs-Str holds pcs-power P = pcs-general-power (pcs-coherent-power P);

registration
let P be pcs-Str ;
cluster pcs-power P ->  strict ;
coherence
 pcs-power P is strict ;
end;

registration
let P be pcs-Str ;
cluster pcs-power P ->  non empty ;
coherence
 not pcs-power P is empty ;
end;

registration
let P be reflexive pcs-Str ;
cluster pcs-power P ->  reflexive ;
coherence
 pcs-power P is reflexive ;
end;

registration
let P be transitive pcs-Str ;
cluster pcs-power P ->  transitive ;
coherence
 pcs-power P is transitive ;
end;

registration
let P be pcs-tol-reflexive pcs-Str ;
cluster pcs-power P ->  pcs-tol-reflexive ;
coherence
 pcs-power P is pcs-tol-reflexive ;
end;

registration
let P be pcs-tol-symmetric pcs-Str ;
cluster pcs-power P ->  pcs-tol-symmetric ;
coherence
 pcs-power P is pcs-tol-symmetric ;
end;

registration
let P be pcs-compatible pcs-Str ;
cluster pcs-power P ->  pcs-compatible ;
coherence
 pcs-power P is pcs-compatible ;
end;

registration
let P be pcs;
cluster pcs-power P ->  pcs-like ;
coherence
 pcs-power P is pcs-like ;
end;