let a9, b9 be Element of REAL ; :: thesis: for a, b being real number st a9 = a & b9 = b holds
* a9,b9 = a * b
let a, b be real number ; :: thesis: ( a9 = a & b9 = b implies * a9,b9 = a * b )
assume that
A1: a9 = a and
A2: b9 = b ; :: thesis: * a9,b9 = a * b
consider x1, x2, y1, y2 being Element of REAL such that
A3: a = [*x1,x2*] and
A4: b = [*y1,y2*] and
A5: a * b = [*(+ (* x1,y1),(opp (* x2,y2))),(+ (* x1,y2),(* x2,y1))*] by XCMPLX_0:def 5;
A6: b = y1 by A4, Lm2;
x2 = 0 by A3, Lm2;
then A7: * x2,y1 = 0 by ARYTM_0:14;
A8: y2 = 0 by A4, Lm2;
then * x1,y2 = 0 by ARYTM_0:14;
then A9: + (* x1,y2),(* x2,y1) = 0 by A7, ARYTM_0:13;
a = x1 by A3, Lm2;
hence * a9,b9 = + (* x1,y1),(* (opp x2),y2) by A1, A2, A6, A8, ARYTM_0:13, ARYTM_0:14
.= + (* x1,y1),(opp (* x2,y2)) by ARYTM_0:17
.= a * b by A5, A9, ARYTM_0:def 7 ;
:: thesis: verum