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)

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

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

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

per cases ( ((chi (X,C)) . c) * ((chi (Y,C)) . c) = 0 or ((chi (X,C)) . c) * ((chi (Y,C)) . c) <> 0 ) ;

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

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

per cases ( (chi (X,C)) . c = 0 or (chi (Y,C)) . c = 0 ) by A2;

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

then not c in X by Th61;
then not c in X /\ Y by XBOOLE_0:def 4;
hence ((chi (X,C)) . c) * ((chi (Y,C)) . c) = (chi ((X /\ Y),C)) . c by A2, Th61; :: thesis: verum

end;

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

then not c in Y by Th61;
then not c in X /\ Y by XBOOLE_0:def 4;
hence ((chi (X,C)) . c) * ((chi (Y,C)) . c) = (chi ((X /\ Y),C)) . c by A2, Th61; :: thesis: verum

end;

end;

end;

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

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;

end;

end;

hence ((chi (X,C)) (#) (chi (Y,C))) . c = (chi ((X /\ Y),C)) . c by VALUED_1:5; :: thesis: verum

end;

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