let X, Y be set ; :: thesis: for C being non empty set holds (chi (X,C)) (#) (chi (Y,C)) = chi ((X /\ Y),C)

let C be non empty set ; :: thesis: (chi (X,C)) (#) (chi (Y,C)) = chi ((X /\ Y),C)

.= C /\ (dom (chi (Y,C))) by Th61

.= C /\ C by Th61

.= dom (chi ((X /\ Y),C)) by Th61 ;

hence (chi (X,C)) (#) (chi (Y,C)) = chi ((X /\ Y),C) by A1, PARTFUN1:5; :: thesis: verum

let C be non empty set ; :: thesis: (chi (X,C)) (#) (chi (Y,C)) = chi ((X /\ Y),C)

A1: now :: thesis: for c being Element of C st c in dom ((chi (X,C)) (#) (chi (Y,C))) holds

((chi (X,C)) (#) (chi (Y,C))) . c = (chi ((X /\ Y),C)) . c

dom ((chi (X,C)) (#) (chi (Y,C))) =
(dom (chi (X,C))) /\ (dom (chi (Y,C)))
by VALUED_1:def 4
((chi (X,C)) (#) (chi (Y,C))) . c = (chi ((X /\ Y),C)) . c

let c be Element of C; :: thesis: ( c in dom ((chi (X,C)) (#) (chi (Y,C))) implies ((chi (X,C)) (#) (chi (Y,C))) . c = (chi ((X /\ Y),C)) . c )

assume c in dom ((chi (X,C)) (#) (chi (Y,C))) ; :: thesis: ((chi (X,C)) (#) (chi (Y,C))) . c = (chi ((X /\ Y),C)) . c

end;assume c in dom ((chi (X,C)) (#) (chi (Y,C))) ; :: thesis: ((chi (X,C)) (#) (chi (Y,C))) . c = (chi ((X /\ Y),C)) . c

now :: thesis: ((chi (X,C)) . c) * ((chi (Y,C)) . c) = (chi ((X /\ Y),C)) . cend;

hence
((chi (X,C)) (#) (chi (Y,C))) . c = (chi ((X /\ Y),C)) . c
by VALUED_1:5; :: thesis: verumper cases
( ((chi (X,C)) . c) * ((chi (Y,C)) . c) = 0 or ((chi (X,C)) . c) * ((chi (Y,C)) . c) <> 0 )
;

end;

suppose A2:
((chi (X,C)) . c) * ((chi (Y,C)) . c) = 0
; :: thesis: ((chi (X,C)) . c) * ((chi (Y,C)) . c) = (chi ((X /\ Y),C)) . c

end;

now :: thesis: ((chi (X,C)) . c) * ((chi (Y,C)) . c) = (chi ((X /\ Y),C)) . cend;

hence
((chi (X,C)) . c) * ((chi (Y,C)) . c) = (chi ((X /\ Y),C)) . c
; :: thesis: verumper cases
( (chi (X,C)) . c = 0 or (chi (Y,C)) . c = 0 )
by A2;

end;

suppose A3:
((chi (X,C)) . c) * ((chi (Y,C)) . c) <> 0
; :: thesis: ((chi (X,C)) . c) * ((chi (Y,C)) . c) = (chi ((X /\ Y),C)) . c

then A4:
(chi (Y,C)) . c <> 0
;

then A5: (chi (Y,C)) . c = 1 by Th67;

A6: c in Y by A4, Th61;

A7: (chi (X,C)) . c <> 0 by A3;

then c in X by Th61;

then A8: c in X /\ Y by A6, XBOOLE_0:def 4;

(chi (X,C)) . c = 1 by A7, Th67;

hence ((chi (X,C)) . c) * ((chi (Y,C)) . c) = (chi ((X /\ Y),C)) . c by A5, A8, Th61; :: thesis: verum

end;then A5: (chi (Y,C)) . c = 1 by Th67;

A6: c in Y by A4, Th61;

A7: (chi (X,C)) . c <> 0 by A3;

then c in X by Th61;

then A8: c in X /\ Y by A6, XBOOLE_0:def 4;

(chi (X,C)) . c = 1 by A7, Th67;

hence ((chi (X,C)) . c) * ((chi (Y,C)) . c) = (chi ((X /\ Y),C)) . c by A5, A8, Th61; :: thesis: verum

.= C /\ (dom (chi (Y,C))) by Th61

.= C /\ C by Th61

.= dom (chi ((X /\ Y),C)) by Th61 ;

hence (chi (X,C)) (#) (chi (Y,C)) = chi ((X /\ Y),C) by A1, PARTFUN1:5; :: thesis: verum