let i be Nat; :: thesis: for R1, R, R2 being Element of i -tuples_on REAL st R1 + R = R2 + R holds
R1 = R2
let R1, R, R2 be Element of i -tuples_on REAL ; :: thesis: ( R1 + R = R2 + R implies R1 = R2 )
assume R1 + R = R2 + R ; :: thesis: R1 = R2
then R1 + (R + (- R)) = (R2 + R) + (- R) by FINSEQOP:29;
then ( R1 + (R + (- R)) = R2 + (R + (- R)) & R + (- R) = i |-> 0 ) by Th22, Th23, BINOP_2:2, FINSEQOP:29, FINSEQOP:77;
then R1 = R2 + (i |-> 0 ) by BINOP_2:2, FINSEQOP:57;
hence R1 = R2 by BINOP_2:2, FINSEQOP:57; :: thesis: verum