let c1, c2 be Number ; ( ex x1, x2, x3, x4, y1, y2, y3, y4 being Real st
( x = [*x1,x2,x3,x4*] & y = [*y1,y2,y3,y4*] & c1 = [*(x1 + y1),(x2 + y2),(x3 + y3),(x4 + y4)*] ) & ex x1, x2, x3, x4, y1, y2, y3, y4 being Real st
( x = [*x1,x2,x3,x4*] & y = [*y1,y2,y3,y4*] & c2 = [*(x1 + y1),(x2 + y2),(x3 + y3),(x4 + y4)*] ) implies c1 = c2 )
given x1, x2, x3, x4, y1, y2, y3, y4 being Real such that A3:
x = [*x1,x2,x3,x4*]
and
A4:
y = [*y1,y2,y3,y4*]
and
A5:
c1 = [*(x1 + y1),(x2 + y2),(x3 + y3),(x4 + y4)*]
; ( for x1, x2, x3, x4, y1, y2, y3, y4 being Real holds
( not x = [*x1,x2,x3,x4*] or not y = [*y1,y2,y3,y4*] or not c2 = [*(x1 + y1),(x2 + y2),(x3 + y3),(x4 + y4)*] ) or c1 = c2 )
given x19, x29, x39, x49, y19, y29, y39, y49 being Real such that A6:
x = [*x19,x29,x39,x49*]
and
A7:
y = [*y19,y29,y39,y49*]
and
A8:
c2 = [*(x19 + y19),(x29 + y29),(x39 + y39),(x49 + y49)*]
; c1 = c2
A9:
x1 = x19
by A3, A6, Th5;
A10:
x2 = x29
by A3, A6, Th5;
A11:
x3 = x39
by A3, A6, Th5;
A12:
x4 = x49
by A3, A6, Th5;
A13:
y1 = y19
by A4, A7, Th5;
A14:
y2 = y29
by A4, A7, Th5;
y3 = y39
by A4, A7, Th5;
hence
c1 = c2
by A4, A5, A7, A8, A9, A10, A11, A12, A13, A14, Th5; verum