let y, z be Element of X3; :: thesis: ( ( for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
y = x3 ) & ( for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
z = x3 ) implies y = z )
assume A11: for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
y = x3 ; :: thesis: ( ex x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st
( x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] & not z = x3 ) or y = z )
assume A12: for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] 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, xx6 being Element of X6, xx7 being Element of X7, xx8 being Element of X8, xx9 being Element of X9 such that
A13: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] by A1, Th180;
y = xx3 by A13, A11;
hence y = z by A13, A12; :: thesis: verum