let R be non empty doubleLoopStr ; :: thesis: for x, y being Scalar of R st x is generated_from_squares & y is being_an_amalgam_of_squares holds
x + y is generated_from_squares
let x, y be Scalar of R; :: thesis: ( x is generated_from_squares & y is being_an_amalgam_of_squares implies x + y is generated_from_squares )
assume that
A1: x is generated_from_squares and
A2: y is being_an_amalgam_of_squares ; :: thesis: x + y is generated_from_squares
consider f being FinSequence of R such that
A3: ( f is being_a_generation_from_squares & x = f /. (len f) ) by A1;
take g = (f ^ <*y*>) ^ <*((f /. (len f)) + y)*>; :: according to O_RING_1:def 14 :: thesis: ( g is being_a_generation_from_squares & x + y = g /. (len g) )
len g = (len (f ^ <*y*>)) + (len <*((f /. (len f)) + y)*>) by FINSEQ_1:22
.= (len (f ^ <*y*>)) + 1 by Lm2 ;
hence ( g is being_a_generation_from_squares & x + y = g /. (len g) ) by A2, A3, Lm3, Lm42; :: thesis: verum