theorem Th2: :: PARSP_2:2

for F being Field
for a, b, c, d being Element of (MPS F) holds
( a,b '||' c,d iff ex e, f, g, h being Element of [: the carrier of F, the carrier of F, the carrier of F:] st
( [[a,b],[c,d]] = [[e,f],[g,h]] & ( ex K being Element of F st
( K * ((e `1_3) - (f `1_3)) = (g `1_3) - (h `1_3) & K * ((e `2_3) - (f `2_3)) = (g `2_3) - (h `2_3) & K * ((e `3_3) - (f `3_3)) = (g `3_3) - (h `3_3) ) or ( (e `1_3) - (f `1_3) = 0. F & (e `2_3) - (f `2_3) = 0. F & (e `3_3) - (f `3_3) = 0. F ) ) ) )