let X1, X2, X3, X4 be set ; :: thesis: ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} implies for x being Element of [:X1,X2,X3,X4:] ex xx1 being Element of X1 ex xx2 being Element of X2 ex xx3 being Element of X3 ex xx4 being Element of X4 st x = [xx1,xx2,xx3,xx4] )
assume that
A1: ( X1 <> {} & X2 <> {} & X3 <> {} ) and
A2: X4 <> {} ; :: thesis: for x being Element of [:X1,X2,X3,X4:] ex xx1 being Element of X1 ex xx2 being Element of X2 ex xx3 being Element of X3 ex xx4 being Element of X4 st x = [xx1,xx2,xx3,xx4]
let x be Element of [:X1,X2,X3,X4:]; :: thesis: ex xx1 being Element of X1 ex xx2 being Element of X2 ex xx3 being Element of X3 ex xx4 being Element of X4 st x = [xx1,xx2,xx3,xx4]
reconsider x9 = x as Element of [:[:X1,X2,X3:],X4:] by ZFMISC_1:def 4;
consider x123 being Element of [:X1,X2,X3:], xx4 being Element of X4 such that
A3: x9 = [x123,xx4] by A2, Lm1, A1;
consider xx1 being Element of X1, xx2 being Element of X2, xx3 being Element of X3 such that
A4: x123 = [xx1,xx2,xx3] by A1, Lm2;
take xx1 ; :: thesis: ex xx2 being Element of X2 ex xx3 being Element of X3 ex xx4 being Element of X4 st x = [xx1,xx2,xx3,xx4]
take xx2 ; :: thesis: ex xx3 being Element of X3 ex xx4 being Element of X4 st x = [xx1,xx2,xx3,xx4]
take xx3 ; :: thesis: ex xx4 being Element of X4 st x = [xx1,xx2,xx3,xx4]
take xx4 ; :: thesis: x = [xx1,xx2,xx3,xx4]
thus x = [xx1,xx2,xx3,xx4] by A3, A4; :: thesis: verum