let K be non empty doubleLoopStr ; :: thesis: for x, y, z, u being Scalar of K
for a, b, c, d being Scalar of (opp K) st x = a & y = b & z = c & u = d holds
( (x + y) + z = (a + b) + c & x + (y + z) = a + (b + c) & (x * y) * z = c * (b * a) & x * (y * z) = (c * b) * a & x * (y + z) = (b + c) * a & (y + z) * x = a * (b + c) & (x * y) + (z * u) = (b * a) + (d * c) )

let x, y, z, u be Scalar of K; :: thesis: for a, b, c, d being Scalar of (opp K) st x = a & y = b & z = c & u = d holds
( (x + y) + z = (a + b) + c & x + (y + z) = a + (b + c) & (x * y) * z = c * (b * a) & x * (y * z) = (c * b) * a & x * (y + z) = (b + c) * a & (y + z) * x = a * (b + c) & (x * y) + (z * u) = (b * a) + (d * c) )

let a, b, c, d be Scalar of (opp K); :: thesis: ( x = a & y = b & z = c & u = d implies ( (x + y) + z = (a + b) + c & x + (y + z) = a + (b + c) & (x * y) * z = c * (b * a) & x * (y * z) = (c * b) * a & x * (y + z) = (b + c) * a & (y + z) * x = a * (b + c) & (x * y) + (z * u) = (b * a) + (d * c) ) )
assume that
A1: ( x = a & y = b & z = c ) and
A2: u = d ; :: thesis: ( (x + y) + z = (a + b) + c & x + (y + z) = a + (b + c) & (x * y) * z = c * (b * a) & x * (y * z) = (c * b) * a & x * (y + z) = (b + c) * a & (y + z) * x = a * (b + c) & (x * y) + (z * u) = (b * a) + (d * c) )
( x * y = b * a & y * z = c * b ) by ;
hence ( (x + y) + z = (a + b) + c & x + (y + z) = a + (b + c) & (x * y) * z = c * (b * a) & x * (y * z) = (c * b) * a & x * (y + z) = (b + c) * a & (y + z) * x = a * (b + c) & (x * y) + (z * u) = (b * a) + (d * c) ) by ; :: thesis: verum