let x, y, z be Quaternion; ( z = x * y implies ( Rea z = ((((Rea x) * (Rea y)) - ((Im1 x) * (Im1 y))) - ((Im2 x) * (Im2 y))) - ((Im3 x) * (Im3 y)) & Im1 z = ((((Rea x) * (Im1 y)) + ((Im1 x) * (Rea y))) + ((Im2 x) * (Im3 y))) - ((Im3 x) * (Im2 y)) & Im2 z = ((((Rea x) * (Im2 y)) + ((Im2 x) * (Rea y))) + ((Im3 x) * (Im1 y))) - ((Im1 x) * (Im3 y)) & Im3 z = ((((Rea x) * (Im3 y)) + ((Im3 x) * (Rea y))) + ((Im1 x) * (Im2 y))) - ((Im2 x) * (Im1 y)) ) )
assume A1:
z = x * y
; ( Rea z = ((((Rea x) * (Rea y)) - ((Im1 x) * (Im1 y))) - ((Im2 x) * (Im2 y))) - ((Im3 x) * (Im3 y)) & Im1 z = ((((Rea x) * (Im1 y)) + ((Im1 x) * (Rea y))) + ((Im2 x) * (Im3 y))) - ((Im3 x) * (Im2 y)) & Im2 z = ((((Rea x) * (Im2 y)) + ((Im2 x) * (Rea y))) + ((Im3 x) * (Im1 y))) - ((Im1 x) * (Im3 y)) & Im3 z = ((((Rea x) * (Im3 y)) + ((Im3 x) * (Rea y))) + ((Im1 x) * (Im2 y))) - ((Im2 x) * (Im1 y)) )
consider x1, x2, x3, x4, y1, y2, y3, y4 being Real such that
A2:
x = [*x1,x2,x3,x4*]
and
A3:
y = [*y1,y2,y3,y4*]
and
A4:
x * y = [*((((x1 * y1) - (x2 * y2)) - (x3 * y3)) - (x4 * y4)),((((x1 * y2) + (x2 * y1)) + (x3 * y4)) - (x4 * y3)),((((x1 * y3) + (y1 * x3)) + (y2 * x4)) - (y4 * x2)),((((x1 * y4) + (x4 * y1)) + (x2 * y3)) - (x3 * y2))*]
by Def9;
A5:
Rea x = x1
by A2, Th16;
A6:
Rea y = y1
by A3, Th16;
A7:
Im1 x = x2
by A2, Th16;
A8:
Im1 y = y2
by A3, Th16;
A9:
Im2 x = x3
by A2, Th16;
A10:
Im2 y = y3
by A3, Th16;
A11:
Im3 x = x4
by A2, Th16;
Im3 y = y4
by A3, Th16;
hence
( Rea z = ((((Rea x) * (Rea y)) - ((Im1 x) * (Im1 y))) - ((Im2 x) * (Im2 y))) - ((Im3 x) * (Im3 y)) & Im1 z = ((((Rea x) * (Im1 y)) + ((Im1 x) * (Rea y))) + ((Im2 x) * (Im3 y))) - ((Im3 x) * (Im2 y)) & Im2 z = ((((Rea x) * (Im2 y)) + ((Im2 x) * (Rea y))) + ((Im3 x) * (Im1 y))) - ((Im1 x) * (Im3 y)) & Im3 z = ((((Rea x) * (Im3 y)) + ((Im3 x) * (Rea y))) + ((Im1 x) * (Im2 y))) - ((Im2 x) * (Im1 y)) )
by A1, A4, A5, A6, A7, A8, A9, A10, A11, Th16; verum