let R be non empty doubleLoopStr ; :: thesis: for x, y being Scalar of R st x is being_a_sum_of_products_of_squares & y is being_a_product_of_squares holds
x + y is being_a_sum_of_products_of_squares
let x, y be Scalar of R; :: thesis: ( x is being_a_sum_of_products_of_squares & y is being_a_product_of_squares implies x + y is being_a_sum_of_products_of_squares )
assume that
A1: x is being_a_sum_of_products_of_squares and
A2: y is being_a_product_of_squares ; :: thesis: x + y is being_a_sum_of_products_of_squares
consider f being FinSequence of R such that
A3: ( f is being_a_Sum_of_products_of_squares & x = f /. (len f) ) by A1;
take g = f ^ <*(x + y)*>; :: according to O_RING_1:def 8 :: thesis: ( g is being_a_Sum_of_products_of_squares & x + y = g /. (len g) )
len g = (len f) + (len <*(x + y)*>) by FINSEQ_1:22
.= (len f) + 1 by Lm2 ;
hence ( g is being_a_Sum_of_products_of_squares & x + y = g /. (len g) ) by A2, A3, Lm3, Lm57; :: thesis: verum