let z1, z2 be set ; :: thesis: ( ( for y1, y2, y3, y4, y5 being set st x = [y1,y2,y3,y4,y5] holds
z1 = y5 ) & ( for y1, y2, y3, y4, y5 being set st x = [y1,y2,y3,y4,y5] holds
z2 = y5 ) implies z1 = z2 )
assume that
A10: for y1, y2, y3, y4, y5 being set st x = [y1,y2,y3,y4,y5] holds
z1 = y5 and
A11: for y1, y2, y3, y4, y5 being set st x = [y1,y2,y3,y4,y5] holds
z2 = y5 ; :: thesis: z1 = z2
z1 = x5 by A1, A10;
hence z1 = z2 by A1, A11; :: thesis: verum