deffunc H_{1}( set ) -> set = ($1 `1) /\ ($1 `2);

let T be set ; :: thesis: for F, G being Subset-Family of T holds card (INTERSECTION (F,G)) c= card [:F,G:]

let F, G be Subset-Family of T; :: thesis: card (INTERSECTION (F,G)) c= card [:F,G:]

set XX = [:F,G:];

set IN = { H_{1}(X) where X is Element of [:(bool T),(bool T):] : X in [:F,G:] } ;

set A = [:(bool T),(bool T):];

set AL = F;

set BL = G;

set C = INTERSECTION (F,G);

A1: { H_{1}(X) where X is Element of [:(bool T),(bool T):] : X in [:F,G:] } c= INTERSECTION (F,G)
_{1}(X) where X is Element of [:(bool T),(bool T):] : X in [:F,G:] }
_{1}(w) where w is Element of [:(bool T),(bool T):] : w in [:F,G:] } c= card [:F,G:]
from TREES_2:sch 2();

hence card (INTERSECTION (F,G)) c= card [:F,G:] by A1, A4, XBOOLE_0:def 10; :: thesis: verum

let T be set ; :: thesis: for F, G being Subset-Family of T holds card (INTERSECTION (F,G)) c= card [:F,G:]

let F, G be Subset-Family of T; :: thesis: card (INTERSECTION (F,G)) c= card [:F,G:]

set XX = [:F,G:];

set IN = { H

set A = [:(bool T),(bool T):];

set AL = F;

set BL = G;

set C = INTERSECTION (F,G);

A1: { H

proof

A4:
INTERSECTION (F,G) c= { H
let a be object ; :: according to TARSKI:def 3 :: thesis: ( not a in { H_{1}(X) where X is Element of [:(bool T),(bool T):] : X in [:F,G:] } or a in INTERSECTION (F,G) )

assume a in { H_{1}(X) where X is Element of [:(bool T),(bool T):] : X in [:F,G:] }
; :: thesis: a in INTERSECTION (F,G)

then consider X being Element of [:(bool T),(bool T):] such that

A2: a = H_{1}(X)
and

A3: X in [:F,G:] ;

( X `1 in F & X `2 in G ) by A3, MCART_1:10;

hence a in INTERSECTION (F,G) by A2, SETFAM_1:def 5; :: thesis: verum

end;assume a in { H

then consider X being Element of [:(bool T),(bool T):] such that

A2: a = H

A3: X in [:F,G:] ;

( X `1 in F & X `2 in G ) by A3, MCART_1:10;

hence a in INTERSECTION (F,G) by A2, SETFAM_1:def 5; :: thesis: verum

proof

card { H
let a be object ; :: according to TARSKI:def 3 :: thesis: ( not a in INTERSECTION (F,G) or a in { H_{1}(X) where X is Element of [:(bool T),(bool T):] : X in [:F,G:] } )

assume a in INTERSECTION (F,G) ; :: thesis: a in { H_{1}(X) where X is Element of [:(bool T),(bool T):] : X in [:F,G:] }

then consider X, Y being set such that

A5: ( X in F & Y in G ) and

A6: a = X /\ Y by SETFAM_1:def 5;

reconsider X = X, Y = Y as Subset of T by A5;

set XY = [X,Y];

A7: ( [X,Y] `1 = X & [X,Y] `2 = Y ) ;

[X,Y] in [:F,G:] by A5, ZFMISC_1:87;

hence a in { H_{1}(X) where X is Element of [:(bool T),(bool T):] : X in [:F,G:] }
by A6, A7; :: thesis: verum

end;assume a in INTERSECTION (F,G) ; :: thesis: a in { H

then consider X, Y being set such that

A5: ( X in F & Y in G ) and

A6: a = X /\ Y by SETFAM_1:def 5;

reconsider X = X, Y = Y as Subset of T by A5;

set XY = [X,Y];

A7: ( [X,Y] `1 = X & [X,Y] `2 = Y ) ;

[X,Y] in [:F,G:] by A5, ZFMISC_1:87;

hence a in { H

hence card (INTERSECTION (F,G)) c= card [:F,G:] by A1, A4, XBOOLE_0:def 10; :: thesis: verum