let z1, z2 be set ; :: thesis: ( ( for y1, y2, y3, y4, y5 being set st x = [y1,y2,y3,y4,y5] holds
z1 = y1 ) & ( for y1, y2, y3, y4, y5 being set st x = [y1,y2,y3,y4,y5] holds
z2 = y1 ) implies z1 = z2 )
assume that
A2: for y1, y2, y3, y4, y5 being set st x = [y1,y2,y3,y4,y5] holds
z1 = y1 and
A3: for y1, y2, y3, y4, y5 being set st x = [y1,y2,y3,y4,y5] holds
z2 = y1 ; :: thesis: z1 = z2
z1 = x1 by A1, A2;
hence z1 = z2 by A1, A3; :: thesis: verum