let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for A, B, C, D, E, F being a_partition of Y
for z, u being Element of Y
for h being Function st G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds
EqClass u,((((B '/\' C) '/\' D) '/\' E) '/\' F) meets EqClass z,A
let G be Subset of (PARTITIONS Y); :: thesis: for A, B, C, D, E, F being a_partition of Y
for z, u being Element of Y
for h being Function st G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds
EqClass u,((((B '/\' C) '/\' D) '/\' E) '/\' F) meets EqClass z,A
let A, B, C, D, E, F be a_partition of Y; :: thesis: for z, u being Element of Y
for h being Function st G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds
EqClass u,((((B '/\' C) '/\' D) '/\' E) '/\' F) meets EqClass z,A
let z, u be Element of Y; :: thesis: for h being Function st G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds
EqClass u,((((B '/\' C) '/\' D) '/\' E) '/\' F) meets EqClass z,A
let h be Function; :: thesis: ( G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F implies EqClass u,((((B '/\' C) '/\' D) '/\' E) '/\' F) meets EqClass z,A )
assume A1: ( G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F ) ; :: thesis: EqClass u,((((B '/\' C) '/\' D) '/\' E) '/\' F) meets EqClass z,A
set GG = EqClass u,((((B '/\' C) '/\' D) '/\' E) '/\' F);
set h = (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A));
A2: dom ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) = G by A1, Th41;
A3: ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . A = EqClass z,A by A1, Th40;
A4: ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . B = EqClass u,B by A1, Th40;
A5: ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . C = EqClass u,C by A1, Th40;
A6: ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . D = EqClass u,D by A1, Th40;
A7: ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . E = EqClass u,E by A1, Th40;
A8: ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . F = EqClass u,F by A1, Th40;
A9: for d being set st d in G holds
((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . d in d

proof

let d be set ; :: thesis: ( d in G implies ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . d in d )
assume A10: d in G ; :: thesis: ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . d in d

now

per cases ( d = A or d = B or d = C or d = D or d = E or d = F ) by A1, A10, ENUMSET1:def 4;

case d = A ; :: thesis: ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . d in d

hence ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . d in d by A3; :: thesis: verum

end;

case d = B ; :: thesis: ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . d in d

hence ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . d in d by A4; :: thesis: verum

end;

case d = C ; :: thesis: ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . d in d

hence ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . d in d by A5; :: thesis: verum

end;

case d = D ; :: thesis: ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . d in d

hence ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . d in d by A6; :: thesis: verum

end;

case d = E ; :: thesis: ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . d in d

hence ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . d in d by A7; :: thesis: verum

end;

case d = F ; :: thesis: ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . d in d

hence ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . d in d by A8; :: thesis: verum

end;

end;

end;

hence ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . d in d ; :: thesis: verum

end;

A in dom ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) by A1, A2, ENUMSET1:def 4;
then A11: ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . A in rng ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) by FUNCT_1:def 5;
B in dom ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) by A1, A2, ENUMSET1:def 4;
then A12: ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . B in rng ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) by FUNCT_1:def 5;
C in dom ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) by A1, A2, ENUMSET1:def 4;
then A13: ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . C in rng ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) by FUNCT_1:def 5;
D in dom ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) by A1, A2, ENUMSET1:def 4;
then A14: ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . D in rng ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) by FUNCT_1:def 5;
E in dom ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) by A1, A2, ENUMSET1:def 4;
then A15: ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . E in rng ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) by FUNCT_1:def 5;
F in dom ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) by A1, A2, ENUMSET1:def 4;
then A16: ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . F in rng ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) by FUNCT_1:def 5;
A17: rng ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) = {(((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . A),(((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . B),(((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . C),(((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . D),(((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . E),(((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . F)} by A1, Th42;
rng ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) c= bool Y

proof

let t be set ; :: according to TARSKI:def 3 :: thesis: ( not t in rng ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) or t in bool Y )
assume A18: t in rng ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) ; :: thesis: t in bool Y

now

per cases ( t = ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . A or t = ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . B or t = ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . C or t = ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . D or t = ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . E or t = ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . F ) by A17, A18, ENUMSET1:def 4;

case t = ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . A ; :: thesis: t in bool Y

hence t in bool Y by A3; :: thesis: verum

end;

case t = ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . B ; :: thesis: t in bool Y

hence t in bool Y by A4; :: thesis: verum

end;

case t = ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . C ; :: thesis: t in bool Y

hence t in bool Y by A5; :: thesis: verum

end;

case t = ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . D ; :: thesis: t in bool Y

hence t in bool Y by A6; :: thesis: verum

end;

case t = ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . E ; :: thesis: t in bool Y

hence t in bool Y by A7; :: thesis: verum

end;

case t = ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) . F ; :: thesis: t in bool Y

hence t in bool Y by A8; :: thesis: verum

end;

end;

end;

hence t in bool Y ; :: thesis: verum

end;

then reconsider FF = rng ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A))) as Subset-Family of Y ;
A19: Intersect FF = meet (rng ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (A .--> (EqClass z,A)))) by A11, SETFAM_1:def 10;
Intersect FF <> {} by A1, A2, A9, BVFUNC_2:def 5;
then consider m being set such that
A20: m in Intersect FF by XBOOLE_0:def 1;
A21: ( m in EqClass z,A & m in EqClass u,B & m in EqClass u,C & m in EqClass u,D & m in EqClass u,E & m in EqClass u,F ) by A3, A4, A5, A6, A7, A8, A11, A12, A13, A14, A15, A16, A19, A20, SETFAM_1:def 1;
EqClass u,((((B '/\' C) '/\' D) '/\' E) '/\' F) = (EqClass u,(((B '/\' C) '/\' D) '/\' E)) /\ (EqClass u,F) by Th1;
then EqClass u,((((B '/\' C) '/\' D) '/\' E) '/\' F) = ((EqClass u,((B '/\' C) '/\' D)) /\ (EqClass u,E)) /\ (EqClass u,F) by Th1;
then EqClass u,((((B '/\' C) '/\' D) '/\' E) '/\' F) = (((EqClass u,(B '/\' C)) /\ (EqClass u,D)) /\ (EqClass u,E)) /\ (EqClass u,F) by Th1;
then A22: (EqClass u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) /\ (EqClass z,A) = (((((EqClass u,B) /\ (EqClass u,C)) /\ (EqClass u,D)) /\ (EqClass u,E)) /\ (EqClass u,F)) /\ (EqClass z,A) by Th1;
m in (EqClass u,B) /\ (EqClass u,C) by A21, XBOOLE_0:def 4;
then m in ((EqClass u,B) /\ (EqClass u,C)) /\ (EqClass u,D) by A21, XBOOLE_0:def 4;
then m in (((EqClass u,B) /\ (EqClass u,C)) /\ (EqClass u,D)) /\ (EqClass u,E) by A21, XBOOLE_0:def 4;
then m in ((((EqClass u,B) /\ (EqClass u,C)) /\ (EqClass u,D)) /\ (EqClass u,E)) /\ (EqClass u,F) by A21, XBOOLE_0:def 4;
then m in (((((EqClass u,B) /\ (EqClass u,C)) /\ (EqClass u,D)) /\ (EqClass u,E)) /\ (EqClass u,F)) /\ (EqClass z,A) by A21, XBOOLE_0:def 4;
hence EqClass u,((((B '/\' C) '/\' D) '/\' E) '/\' F) meets EqClass z,A by A22, XBOOLE_0:def 7; :: thesis: verum