let x', y' be Element of REAL ; :: thesis: for x, y being real number st x' = x & y' = y holds
* x',y' = x * y
let x, y be real number ; :: thesis: ( x' = x & y' = y implies * x',y' = x * y )
assume A1: ( x' = x & y' = y ) ; :: thesis: * x',y' = x * y
consider x1, x2, y1, y2 being Element of REAL such that
A2: x = [*x1,x2*] and
A3: y = [*y1,y2*] and
A4: x * y = [*(+ (* x1,y1),(opp (* x2,y2))),(+ (* x1,y2),(* x2,y1))*] by XCMPLX_0:def 5;
A5: ( x = x1 & y = y1 ) by A2, A3, Lm7;
A6: ( x2 = 0 & y2 = 0 ) by A2, A3, Lm7;
then ( * x1,y2 = 0 & * x2,y1 = 0 ) by ARYTM_0:14;
then A7: + (* x1,y2),(* x2,y1) = 0 by ARYTM_0:13;
thus * x',y' = + (* x1,y1),(* (opp x2),y2) by A1, A5, A6, ARYTM_0:13, ARYTM_0:14
.= + (* x1,y1),(opp (* x2,y2)) by ARYTM_0:17
.= x * y by A4, A7, ARYTM_0:def 7 ; :: thesis: verum