let y, z be Element of X1; :: thesis: ( ( for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
y = x1 ) & ( for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
z = x1 ) implies y = z )
assume A3: for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
y = x1 ; :: thesis: ( ex x1, x2, x3, x4 being set st
( x = [x1,x2,x3,x4] & not z = x1 ) or y = z )
assume A4: for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
z = x1 ; :: thesis: y = z
consider xx1 being Element of X1, xx2 being Element of X2, xx3 being Element of X3, xx4 being Element of X4 such that
A5: x = [xx1,xx2,xx3,xx4] by A1, Lm3;
y = xx1 by A5, A3;
hence y = z by A5, A4; :: thesis: verum