let X1, X2, X3 be set ; :: thesis: ( X1 <> {} & X2 <> {} & X3 <> {} implies for x being Element of [:X1,X2,X3:] ex xx1 being Element of X1 ex xx2 being Element of X2 ex xx3 being Element of X3 st x = [xx1,xx2,xx3] )
assume that
A1: ( X1 <> {} & X2 <> {} ) and
A2: X3 <> {} ; :: thesis: for x being Element of [:X1,X2,X3:] ex xx1 being Element of X1 ex xx2 being Element of X2 ex xx3 being Element of X3 st x = [xx1,xx2,xx3]
let x be Element of [:X1,X2,X3:]; :: thesis: ex xx1 being Element of X1 ex xx2 being Element of X2 ex xx3 being Element of X3 st x = [xx1,xx2,xx3]
reconsider x9 = x as Element of [:[:X1,X2:],X3:] by ZFMISC_1:def 3;
consider x12 being Element of [:X1,X2:], xx3 being Element of X3 such that
A3: x9 = [x12,xx3] by A1, A2, Lm1;
consider xx1 being Element of X1, xx2 being Element of X2 such that
A4: x12 = [xx1,xx2] by A1, Lm1;
take xx1 ; :: thesis: ex xx2 being Element of X2 ex xx3 being Element of X3 st x = [xx1,xx2,xx3]
take xx2 ; :: thesis: ex xx3 being Element of X3 st x = [xx1,xx2,xx3]
take xx3 ; :: thesis: x = [xx1,xx2,xx3]
thus x = [xx1,xx2,xx3] by A3, A4; :: thesis: verum