theorem Th8: :: C0SP1:8

for V being Algebra
for V1 being Subalgebra of V holds
( ( for v1, w1 being Element of V1
for v, w being Element of V st v1 = v & w1 = w holds
v1 + w1 = v + w ) & ( for v1, w1 being Element of V1
for v, w being Element of V st v1 = v & w1 = w holds
v1 * w1 = v * w ) & ( for v1 being Element of V1
for v being Element of V
for a being Real st v1 = v holds
a * v1 = a * v ) & 1_ V1 = 1_ V & 0. V1 = 0. V )