let X1, X2, X3, X4, X5, X6, X7 be set ; :: thesis: ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} implies for x being Element of [:X1,X2,X3,X4,X5,X6,X7:] ex xx1 being Element of X1 ex xx2 being Element of X2 ex xx3 being Element of X3 ex xx4 being Element of X4 ex xx5 being Element of X5 ex xx6 being Element of X6 ex xx7 being Element of X7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] )
assume that
A1: ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} ) and
A2: X7 <> {} ; :: thesis: for x being Element of [:X1,X2,X3,X4,X5,X6,X7:] ex xx1 being Element of X1 ex xx2 being Element of X2 ex xx3 being Element of X3 ex xx4 being Element of X4 ex xx5 being Element of X5 ex xx6 being Element of X6 ex xx7 being Element of X7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7]
let x be Element of [:X1,X2,X3,X4,X5,X6,X7:]; :: 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 ex xx5 being Element of X5 ex xx6 being Element of X6 ex xx7 being Element of X7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7]
reconsider x9 = x as Element of [:[:X1,X2,X3,X4,X5,X6:],X7:] ;
[:X1,X2,X3,X4,X5,X6:] <> {} by A1, Th51;
then consider x123456 being Element of [:X1,X2,X3,X4,X5,X6:], xx7 being Element of X7 such that
A3: x9 = [x123456,xx7] by A2, Lm1;
consider xx1 being Element of X1, xx2 being Element of X2, xx3 being Element of X3, xx4 being Element of X4, xx5 being Element of X5, xx6 being Element of X6 such that
A4: x123456 = [xx1,xx2,xx3,xx4,xx5,xx6] by A1, Th55;
take xx1 ; :: thesis: ex xx2 being Element of X2 ex xx3 being Element of X3 ex xx4 being Element of X4 ex xx5 being Element of X5 ex xx6 being Element of X6 ex xx7 being Element of X7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7]
take xx2 ; :: thesis: ex xx3 being Element of X3 ex xx4 being Element of X4 ex xx5 being Element of X5 ex xx6 being Element of X6 ex xx7 being Element of X7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7]
take xx3 ; :: thesis: ex xx4 being Element of X4 ex xx5 being Element of X5 ex xx6 being Element of X6 ex xx7 being Element of X7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7]
take xx4 ; :: thesis: ex xx5 being Element of X5 ex xx6 being Element of X6 ex xx7 being Element of X7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7]
take xx5 ; :: thesis: ex xx6 being Element of X6 ex xx7 being Element of X7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7]
take xx6 ; :: thesis: ex xx7 being Element of X7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7]
take xx7 ; :: thesis: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7]
thus x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A3, A4; :: thesis: verum