let x9, y9 be Element of REAL ; :: thesis: for x, y being Real st x9 = x & y9 = y holds
* (x9,y9) = x * y
let x, y be Real; :: thesis: ( x9 = x & y9 = y implies * (x9,y9) = x * y )
assume A1: ( x9 = x & y9 = y ) ; :: thesis: * (x9,y9) = 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;
x2 = 0 by A2, Lm7;
then A5: * (x2,y1) = 0 by ARYTM_0:12;
A6: y2 = 0 by A3, Lm7;
then * (x1,y2) = 0 by ARYTM_0:12;
then A7: + ((* (x1,y2)),(* (x2,y1))) = 0 by A5, ARYTM_0:11;
( x = x1 & y = y1 ) by A2, A3, Lm7;
hence * (x9,y9) = + ((* (x1,y1)),(* ((opp x2),y2))) by A1, A6, ARYTM_0:11, ARYTM_0:12
.= + ((* (x1,y1)),(opp (* (x2,y2)))) by ARYTM_0:15
.= x * y by A4, A7, ARYTM_0:def 5 ;
:: thesis: verum