My example is based on all MSCI Index tracking ETFs. Therefore:
The MSCI ACWI IMI (all world, large and small caps included) market cap is currently 50,229,509.24 M$ as per this document:
https://www.msci.com/documents/10199/4211cc4b-453d-4b0a-a6a7-51d36472a703
MSCI World (which is Developed Large/Mid-cap) is 38,521,766.09 M$ as per this document
https://www.msci.com/documents/10199/178e6643-6ae6-47b9-82be-e1fc565ededb
MSCI World Small Cap (Developed Small Cap is 5,844,080.86 M$ as per this document
https://www.msci.com/documents/10199/a67b0d43-0289-4bce-8499-0c102eaa8399
MSCI Emerging Markets IMI (Emerging Large/Mid/Small-Cap) is 5,863,662.29 M$ as per this document
https://www.msci.com/documents/10199/97e25eb7-9bd0-4204-bea9-077095acf1d3
Therefore:
MSCI World is 76.7% of ACWI IMI
MSCI World Small Cap is 11.6% of ACWI IMI
MSCI EM IMI is 11.7% of ACWI IMI
adds up to exactly 100% so the proof is valid 