let y2, X1, X2, X3, X4, X5, X6, X7, X8 be set ; :: thesis: for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3
for xx4 being Element of X4
for xx5 being Element of X5
for xx6 being Element of X6
for xx7 being Element of X7
for xx8 being Element of X8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] holds
y2 = xx2 ) holds
y2 = x `2
let x be Element of [:X1,X2,X3,X4,X5,X6,X7,X8:]; :: thesis: ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3
for xx4 being Element of X4
for xx5 being Element of X5
for xx6 being Element of X6
for xx7 being Element of X7
for xx8 being Element of X8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] holds
y2 = xx2 ) implies y2 = x `2 )
assume that
A1: ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} ) and
A2: for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3
for xx4 being Element of X4
for xx5 being Element of X5
for xx6 being Element of X6
for xx7 being Element of X7
for xx8 being Element of X8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] holds
y2 = xx2 ; :: thesis: y2 = x `2
x = [(x `1),(x `2),(x `3),(x `4),(x `5),(x `6),(x `7),(x `8)] by A1, Th138;
hence y2 = x `2 by A2; :: thesis: verum