let i be natural Number ; for R, R1, R2 being Element of i -tuples_on REAL st R1 + R = R2 + R holds
R1 = R2
let R, R1, R2 be Element of i -tuples_on REAL; ( R1 + R = R2 + R implies R1 = R2 )
assume
R1 + R = R2 + R
; R1 = R2
then
R1 + (R + (- R)) = (R2 + R) + (- R)
by FINSEQOP:28;
then A1:
R1 + (R + (- R)) = R2 + (R + (- R))
by FINSEQOP:28;
R + (- R) = i |-> 0
by Th8, Th9, BINOP_2:2, FINSEQOP:73;
then
R1 = R2 + (i |-> 0)
by A1, BINOP_2:2, FINSEQOP:56;
hence
R1 = R2
by BINOP_2:2, FINSEQOP:56; verum