let X1, X2, X3, y3 be set ; :: thesis: for x being Element of [:X1,X2,X3:] st X1 <> {} & X2 <> {} & X3 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3 st x = [xx1,xx2,xx3] holds
y3 = xx3 ) holds
y3 = x `3
let x be Element of [:X1,X2,X3:]; :: thesis: ( X1 <> {} & X2 <> {} & X3 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3 st x = [xx1,xx2,xx3] holds
y3 = xx3 ) implies y3 = x `3 )
assume that
A1: ( X1 <> {} & X2 <> {} & X3 <> {} ) and
A2: for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3 st x = [xx1,xx2,xx3] holds
y3 = xx3 ; :: thesis: y3 = x `3
x = [(x `1),(x `2),(x `3)] by A1, Th48;
hence y3 = x `3 by A2; :: thesis: verum