let y4, X1, X2, X3, X4, X5 be set ; :: thesis: for x being Element of [:X1,X2,X3,X4,X5:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & ( 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 st x = [xx1,xx2,xx3,xx4,xx5] holds
y4 = xx4 ) holds
y4 = x `4
let x be Element of [:X1,X2,X3,X4,X5:]; :: thesis: ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & ( 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 st x = [xx1,xx2,xx3,xx4,xx5] holds
y4 = xx4 ) implies y4 = x `4 )
assume that
A1: ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} ) 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 st x = [xx1,xx2,xx3,xx4,xx5] holds
y4 = xx4 ; :: thesis: y4 = x `4
x = [(x `1),(x `2),(x `3),(x `4),(x `5)] by A1, Th19;
hence y4 = x `4 by A2; :: thesis: verum