let c1, c2, c3 be Quaternion; :: thesis: (c1 + c2) * c3 = (c1 * c3) + (c2 * c3)
consider x1, y1, w1, z1 being Element of REAL such that
A1: c1 = [*x1,y1,w1,z1*] by Lm1;
consider x2, y2, w2, z2 being Element of REAL such that
A2: c2 = [*x2,y2,w2,z2*] by Lm1;
consider x3, y3, w3, z3 being Element of REAL such that
A3: c3 = [*x3,y3,w3,z3*] by Lm1;
A4: (c1 + c2) * c3 = [*(x1 + x2),(y1 + y2),(w1 + w2),(z1 + z2)*] * c3 by A1, A2, QUATERNI:def 7
.= [*(((((x1 + x2) * x3) - ((y1 + y2) * y3)) - ((w1 + w2) * w3)) - ((z1 + z2) * z3)),(((((x1 + x2) * y3) + ((y1 + y2) * x3)) + ((w1 + w2) * z3)) - ((z1 + z2) * w3)),(((((x1 + x2) * w3) + (x3 * (w1 + w2))) + (y3 * (z1 + z2))) - (z3 * (y1 + y2))),(((((x1 + x2) * z3) + ((z1 + z2) * x3)) + ((y1 + y2) * w3)) - ((w1 + w2) * y3))*] by A3, QUATERNI:def 10
.= [*(((((x1 * x3) + (x2 * x3)) - ((y1 * y3) + (y2 * y3))) - ((w1 * w3) + (w2 * w3))) - ((z1 * z3) + (z2 * z3))),(((((x1 * y3) + (x2 * y3)) + ((y1 * x3) + (y2 * x3))) + ((w1 * z3) + (w2 * z3))) - ((z1 * w3) + (z2 * w3))),(((((x1 * w3) + (x2 * w3)) + ((x3 * w1) + (x3 * w2))) + ((y3 * z1) + (y3 * z2))) - ((z3 * y1) + (z3 * y2))),(((((x1 * z3) + (x2 * z3)) + ((z1 * x3) + (z2 * x3))) + ((y1 * w3) + (y2 * w3))) - ((w1 * y3) + (w2 * y3)))*] ;
(c1 * c3) + (c2 * c3) = [*((((x1 * x3) - (y1 * y3)) - (w1 * w3)) - (z1 * z3)),((((x1 * y3) + (y1 * x3)) + (w1 * z3)) - (z1 * w3)),((((x1 * w3) + (x3 * w1)) + (y3 * z1)) - (z3 * y1)),((((x1 * z3) + (z1 * x3)) + (y1 * w3)) - (w1 * y3))*] + (c2 * c3) by A1, A3, QUATERNI:def 10
.= [*((((x1 * x3) - (y1 * y3)) - (w1 * w3)) - (z1 * z3)),((((x1 * y3) + (y1 * x3)) + (w1 * z3)) - (z1 * w3)),((((x1 * w3) + (x3 * w1)) + (y3 * z1)) - (z3 * y1)),((((x1 * z3) + (z1 * x3)) + (y1 * w3)) - (w1 * y3))*] + [*((((x2 * x3) - (y2 * y3)) - (w2 * w3)) - (z2 * z3)),((((x2 * y3) + (y2 * x3)) + (w2 * z3)) - (z2 * w3)),((((x2 * w3) + (x3 * w2)) + (y3 * z2)) - (z3 * y2)),((((x2 * z3) + (z2 * x3)) + (y2 * w3)) - (w2 * y3))*] by A2, A3, QUATERNI:def 10
.= [*(((((x1 * x3) - (y1 * y3)) - (w1 * w3)) - (z1 * z3)) + ((((x2 * x3) - (y2 * y3)) - (w2 * w3)) - (z2 * z3))),(((((x1 * y3) + (y1 * x3)) + (w1 * z3)) - (z1 * w3)) + ((((x2 * y3) + (y2 * x3)) + (w2 * z3)) - (z2 * w3))),(((((x1 * w3) + (x3 * w1)) + (y3 * z1)) - (z3 * y1)) + ((((x2 * w3) + (x3 * w2)) + (y3 * z2)) - (z3 * y2))),(((((x1 * z3) + (z1 * x3)) + (y1 * w3)) - (w1 * y3)) + ((((x2 * z3) + (z2 * x3)) + (y2 * w3)) - (w2 * y3)))*] by QUATERNI:def 7 ;
hence (c1 + c2) * c3 = (c1 * c3) + (c2 * c3) by A4; :: thesis: verum