let i, j be Nat; :: thesis: for M1, M2 being Matrix of COMPLEX st len M1 = len M2 & width M1 = width M2 & [i,j] in Indices M1 holds
(M1 + M2) * i,j = (M1 * i,j) + (M2 * i,j)
let M1, M2 be Matrix of COMPLEX ; :: thesis: ( len M1 = len M2 & width M1 = width M2 & [i,j] in Indices M1 implies (M1 + M2) * i,j = (M1 * i,j) + (M2 * i,j) )
assume A1: ( len M1 = len M2 & width M1 = width M2 & [i,j] in Indices M1 ) ; :: thesis: (M1 + M2) * i,j = (M1 * i,j) + (M2 * i,j)
A2: COMPLEX2Field (M1 + M2) = COMPLEX2Field (Field2COMPLEX ((COMPLEX2Field M1) + (COMPLEX2Field M2))) by MATRIX_5:def 3
.= (COMPLEX2Field M1) + (COMPLEX2Field M2) by MATRIX_5:6 ;
A3: [i,j] in Indices (COMPLEX2Field M1) by A1, MATRIX_5:def 1;
set m = COMPLEX2Field (M1 + M2);
reconsider m' = COMPLEX2Field (M1 + M2) as Matrix of COMPLEX by COMPLFLD:def 1;
reconsider m1 = COMPLEX2Field M1, m2 = COMPLEX2Field M2 as Matrix of COMPLEX by COMPLFLD:def 1;
A4: M1 * i,j = m1 * i,j by MATRIX_5:def 1
.= (COMPLEX2Field M1) * i,j by COMPLFLD:def 1 ;
A5: M2 * i,j = m2 * i,j by MATRIX_5:def 1
.= (COMPLEX2Field M2) * i,j by COMPLFLD:def 1 ;
(M1 + M2) * i,j = m' * i,j by MATRIX_5:def 1
.= (COMPLEX2Field (M1 + M2)) * i,j by COMPLFLD:def 1
.= ((COMPLEX2Field M1) * i,j) + ((COMPLEX2Field M2) * i,j) by A2, A3, MATRIX_3:def 3 ;
hence (M1 + M2) * i,j = (M1 * i,j) + (M2 * i,j) by A4, A5; :: thesis: verum