let y, z be Element of X3; :: thesis: ( ( for x1, x2, x3, x4, x5 being set st x = [x1,x2,x3,x4,x5] holds
y = x3 ) & ( for x1, x2, x3, x4, x5 being set st x = [x1,x2,x3,x4,x5] holds
z = x3 ) implies y = z )
assume A11: for x1, x2, x3, x4, x5 being set st x = [x1,x2,x3,x4,x5] holds
y = x3 ; :: thesis: ( ex x1, x2, x3, x4, x5 being set st
( x = [x1,x2,x3,x4,x5] & not z = x3 ) or y = z )
assume A12: for x1, x2, x3, x4, x5 being set st x = [x1,x2,x3,x4,x5] holds
z = x3 ; :: thesis: y = z
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 such that
A13: x = [xx1,xx2,xx3,xx4,xx5] by A1, Th17;
y = xx3 by A13, A11;
hence y = z by A13, A12; :: thesis: verum