let a', b' be Element of REAL ; :: thesis: for a, b being real number st a' = a & b' = b holds
+ a',b' = a + b
let a, b be real number ; :: thesis: ( a' = a & b' = b implies + a',b' = a + b )
assume that
A1: a' = a and
A2: b' = b ; :: thesis: + a',b' = 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),(+ x2,y2)*] by XCMPLX_0:def 4;
A6: y2 = 0 by A4, Lm2;
x2 = 0 by A3, Lm2;
then A7: + x2,y2 = 0 by A6, ARYTM_0:13;
A8: b = y1 by A4, Lm2;
a = x1 by A3, Lm2;
hence + a',b' = a + b by A1, A2, A5, A8, A7, ARYTM_0:def 7; :: thesis: verum