let x, y be VECTOR of CLSStruct(# ZS,O,F,G #); reconsider X = x, Y = y as Element of ZS ;
( x + y = H1(X,Y) & y + x = H1(Y,X) )
by A1; hence
x + y = y + x
;
end;
thus
for x, y, z being VECTOR of CLSStruct(# ZS,O,F,G #) holds (x + y)+ z = x +(y + z)
let x, y, z be VECTOR of CLSStruct(# ZS,O,F,G #); reconsider X = x, Y = y, Z = z as Element of ZS ;
( (x + y)+ z = H1(H1(X,Y),Z) & x +(y + z)= H1(X,H1(Y,Z)) )
by A1; hence (x + y)+ z = x +(y + z)
;
let z be Complex; let x, y be VECTOR of CLSStruct(# ZS,O,F,G #); reconsider X = x, Y = y as Element of ZS ; (z * x)+(z * y)= H1(O,O)
by A1; hence
z *(x + y)=(z * x)+(z * y)by A2;
end;
thus
for z1, z2 being Complex for x being VECTOR of CLSStruct(# ZS,O,F,G #) holds (z1 + z2)* x =(z1 * x)+(z2 * x)
let z1, z2 be Complex; let x be VECTOR of CLSStruct(# ZS,O,F,G #); set c = z1 + z2; reconsider X = x as Element of ZS ; (z1 + z2)* x = O
by A2; hence (z1 + z2)* x =(z1 * x)+(z2 * x)by A1;
end;
thus
for z1, z2 being Complex for x being VECTOR of CLSStruct(# ZS,O,F,G #) holds (z1 * z2)* x = z1 *(z2 * x)
let z1, z2 be Complex; let x be VECTOR of CLSStruct(# ZS,O,F,G #); set c = z1 * z2; reconsider X = x as Element of ZS ; (z1 * z2)* x = O
by A2; hence (z1 * z2)* x = z1 *(z2 * x)by A2;